' 



-.. V : 





) 







1. Telescopic view of tke -full Moon.. S.Telogcopic TTIPW of Satiuox Asliis T.-TU.E:?. 

2. io do of a -part of tlio Moon aiear q^uattrature . 4. do of Jupiter &lii.s "Moons. 




INTRODUCTION 



ASTEONOMY; 



DESIGNED AS A 



TEXT BOOK 




STUDENTS OF YALE COLLEGE. 



REVISED EDITION, 

NUMEEOTTS ALTERATIONS AND ADDITIONS, INCLUDING THE LATEST 
DISCOVERIES. 



BY DENISOX OLMSTED, LL.D., 



PROFESSOR OF NATURAL PIIILOSOPET 



NEW YORK: 

ROBERT B. COLLINS, 254 PEARL-STREET. 
1855. 







Entered, according to Act of Congress, in the year 1854, 

By DENISON OLMSTED, 
In the Clerk's Office of the District Court of Connecticut. 



STEREOTYPED BY 

KICHABD c. VALENT'INE, 

17 Dutcii-st., cor. of Fulton. 



REVISED EDITION. 



THE great progress of astronomical discovery, during the *ast few years, has 
induced the author of this work to prepare, with much labor and expense, a 
new edition, which should fully exhibit the features of the science in their 
latest physiognomy. Since the stereotype edition was first published, in 
1844, numerous and important discoveries have been made, both in the solar 
system and among the fixed stars. The dimensions of the planetary system 
have been nearly doubled by the addition of the planet Neptune ; the number 
of the Asteroids has been increased from four to twenty-seven ; interesting 
discoveries have been made in the Rings of Saturn, and an eighth member has 
been added to his retinue of satellites ; and the subject of Comets has received 
a new impulse by the appearance of the remarkable cornet of 1843. 

Meanwhile, the introduction into practical astronomy of Telescopes greater 
exceeding in light and power those previously directed to the heavens, has dis- 
closed new wonders among the fixed stars, and especially among the Nebulso ; 
a problem, which had eluded the eager pursuit of astronomers, that of find- 
ing the distances of the fixed stars, has been solved ; stellar astronomy has 
been greatly enriched by the observations of Sir John Herschel in the southern 
hemisphere ; our knowledge of the proper motions of the stars has been en- 
larged, and crowned with several most interesting results ; some progress has 
been made towards determining the magnitudes of the stars, the nature of 
their orbits, the velocities of their motions, and the periods of their revolu- 
tions ; and recent investigations inspire the hope, that the mechanism of the 
universe will shortly be understood as perfectly as is that of the solar system. 

Besides a notice of these important results, there will also be found, in the 
present edition, a concise statement of the phenomena and causes of Meteoric 
Showers, a topic which, for peculiar reasons, the author has forborne to in- 
troduce into the text of previous editions, but it is now inserted in conformity 
with the example of Sir John Herschel, Humboldt, and other distinguished 
writers of astronomical works, who have not scrupled to assign to the period- 
ical meteors a distinct place in the solar system. Hie Numerical Relations oi 
the sun and planets a subject which has heretofore appeared only in the 
Addenda is now incorporated with the text ; and, illustrated as it is by a 
variety of curious and interesting problems, it will lead the pupil to form just 
and accurate ideas of those relations, and of the laws that govern them. 

The part of astronomy which relates to the Earth, the Sun, and the Moon, 
has undergone, of late, but few changes ; but that which relates to the Planets 
and Fixed Stars has been enriched by so many new discoveries, that it has 
been necessary to re- write it, and to cast it anew. With these improvements, 
it is believed the present work will be found to be as well suited to initiate 
the student of astronomy into the mysteries of this noble science, and to in- 
spire a taste for its pursuits, as can reasonably be expected of any work com- 
prised within such narrow limits. 

YALE COLLEGE, January, 1854. 



PREFACE. 



NEARLY all who have written Treatises on Astronomy, designed for young 
learners, appear to have erred in one of two ways ; they have either disre- 
garded demonstrative evidence, and relied on mere popular illustration, or they 
have exhibited the elements of the science in naked mathematical formulae. 
The former are usually diffuse and superficial ; the latter, technical and ab- 
struse. 

In the following Treatise, we have endeavored to unite the advantages of 
both methods. We have sought, first, to establish the great principles of 
astronomy on a mathematical basis ; and, secondly, to render the study inter- 
esting and intelligible to the learner, by easy and familiar illustrations. We 
would not encourage any one to believe that he can enjoy a full view of the 
grand edifice of astronomy, while its noble foundations are hidden from his 
sight; nor would we assure him that he can contemplate the structure in its 
true magnificence, while its basement alone is within his field of vision. We 
would, therefore, that the student of astronomy should confine his attention 
neither to the exterior of the building, nor to the mere analytic investigation 
of its structure. We would desire that he should not only study it in models 
and diagrams, and mathematical formulae, but should at the same time acquire 
a love of nature herself, and cultivate the habit of raising his views to the 
grand originals. Nor is the effort to form a clear conception of the motions and 
dimensions of the heavenly bodies, less favorable to the improvement of the 
intellectual powers, than the study of pure geometry. 

But it is evidently possible to follow out all the intricacies of an analytical 
process, and to arrive at a full conviction of the great truths of astronomy, and 
yet know very little of nature. According to our experience, however, but few 
students in the course of a liberal education will feel satisfied with this. They 
do not need so much to be convinced that the assertions of astronomers are 
true, as they desire to know what the truths are, and how they were ascer- 
tained ; and they will derive from the study of astronomy little of that moral 
and intellectual elevation ^hich they had anticipated, unless they learn to look 
upon the heavens with new views, and a clear comprehension of their won- 
derful mechanism. 

Much of the difficulty that usually attends the early progress of the astro- 
nomical student, arises from his being too soon introduced to the most perplex- 
ing part of the whole subject, the planetary motions. In this work, the con- ' 
sideration of these is for the most part postponed until the learner has become 
familiar with the artificial circles of the sphere, and conversant with the celes- 
tial bodies. We then first take the most simple view possible of the planetary 
motions by contemplating them as they really are in nature, and afterwards 
proceed to the more difficult inquiry, why they appear as they do. Probably 
no science derives such signal advantage from a happy arrangement, as as- 
tronomy ; an order, which brings out every fact or doctrine of the science just 
in the place where the mind of the learner is prepared to receive it. 



ANALYSIS. 

DESIGNED AS A BASIS FOR REVIEW AND EXAMINATION. 



PRELIMINARY OBSERVATIONS. 



Page 



Astronomy defined, 

Descriptive Astronomy, 

Physical do. 

Practical do. 

History. Ancient nations who cultiva- 
ted astronomy, 

Pythagoras his age and country, 

His views of the celestial motions, 

Alexandrian School when founded 
by whom introduction of astronomi- 
cal instruments, 

Hipparchus his character, 2 

Ptolemy the Almagest, 

Copernicus, Tycho Brahe, Keplqr and 
Galileo respective labors of each,.... 2 

Sir Isaac Newton his great discovery, 2 

La Place Mecanique Celeste, 2 

Astrology Natural and Judicial ob- 
ject of each, 2 

Accuracy aimed at by astronomers, 

Copernican System its leading doc- 
trines, 3 

Plan of this work, 

Part I. OF THE EARTH. 



Chapter 1. OF THE FIGURE AND DIMENSIONS 

OF THE EARTH, AND THE DOCTRINE OF THE 
SPHERE. 

Figure of the earth, 4 

Proofs, 4 

Dip of the horizon, 4 

How found, 5 

Table of the dip its use, 

Exact figure of the earth, 6 

Its circumference, 6 

Small inequalities of the earth's surface, 6 

Diameter of the earth how determined, 7 
How to divest the mind of preconceived 

erroneous notions, J 

DOCTRINE OF THE SPHERE, defined, 9 

Great and small circles defined, 9 

Axis of a circle pole, 9 

Situation of the poles of two great cir- 
cles which cut each other at right an- 

gles, 9 

Points of intersection of two great cir- 
cles how many degrees apart, 10 



Page 

When a great circle passes through the 

pole of another, how does it cut it ? . 10 

Secondary defined, 10 

Angle made by two great circles how 

measured, 10 

Terrestrial and Celestial spheres distin- 
guished, 10 

Horizon defined, 11 

Sensible horizon, 11 

Rational do 11 

Zenith and Nadir, 11 

Vertical circles, 11 

Meridian, 11 

Prime Vertical, 11 

How the place of a celestial body is de- 
termined, 11 

Altitude azimuth amplitude, 12 

Zenith Distance how measured, 12 

Axis of the earth axis of the celestial 

sphere, 12 

Poles of the earth poles of the heav- 
ens, - 

Equator terrestrial and celestial, 12 

3 Hourcircles, 13 

Latitude, " 

Polar Distance, how related to latitude, 13 

Longitude, 13 

Standard Meridians, 13 

Ecliptic, 13 

Inclination of 'the ecliptic to the equa- 
tor, : 

Equinoctial points, 

Equinoxes Vernal and autumnal, .... 13 

6 Solstitial points, 14 

Solstices, 14 

Signs of the ecliptic enumerated, 14 

Colures Equinoctial and Solstitial,... 14 

Right ascension, 15 

Declination, 15 

Celestial Longitude, 15 

Celestial Latitude, 15 

North Polar Distances, how related to 

latitude, 15 

Parallels of Latitude, 15 

Tropics, : 

Polar circles, > 

Zones, 16 

Zodiac, 36 



Yl 



ANALYSIS. 



Page. 



16 



Elevation ojttne pole to what is it 
equal ?.... 

Elevation of the equator, 

Distance of a place from the pole, to 
what equal? 16 



Chapter II. DIURNAL REVOLUTION ARTI- 
FICIAL GLOBES ASTRONOMICAL PROBLEMS. 

Circles of Diurnal Revolution, 17 

Sidereal day defined, < 17 

Appearance of the circles of diurnal 

revolution at the equator, 17 

A Right Sphere defined, 18 

A Parallel Sphere, 19 

An Oblique Sphere, 19 

Circle of Perpetual Apparition, 20 

Circle of Perpetual Occupation, 20 

How are the circles of daily motion cut 
by the horizon in the different 

spheres? 20 

Explanation of the peculiar appearan- 
ces of each sphere, from the revolu- 
tion of the earth on its axis, 21 

Artificial Globes terrestrial and celes- 
tial, 22 

Their use, 23 

Meridian how represented how gra- 
duated, 23 

Horizon how represented how gra- 
duated, 23 

Hour Circles, how represented, 23 

Hour Index described, 23 

Quadrant of Altitude, 24 

Its use described, 24 

To rectify the globe for any place, 24 

PROBLEMS ON THE TERRESTRIAL GLOBE 
To find the latitude and longitude 

of a place, 24 

To find a place, its latitude and longi- 

gitude being given, 25 

To find the bearing and distance of 

two places, 25 

To determine the difference of time of 

two places, 25 

The hour being given at any place, to 
tell what hour it is in any other part 

of the world,... 25 

To find the antoeci, veriaeci, and antipo- 

des, 25 

To rectify the globe for the sun's place, 26 
The latitude of the place being given, 
to find the time of the sun's rising 

and setting, 26 

PROBLEMS ON THE CELESTIAL GLOBE. 
To find the right ascension and decli- 
nation, 26 

To represent the appearance of the 

heavens at any time, 27 

To find the altitude and azimuth of a 
star 27 



To find the angular distance of two 

stars from each other, 27 

16 To find the sun's meridian altitude, 
the latitude and day of the month 
being given,. 28 

Chapter III. PARALLAX REFRACTIO* 
TWILIGHT. 



Page 



Parallax defined, 28 

Horizontal Parallax, 29 

Relation of parallax to the zenith dis- 
tance, and distance from the center 

of the earth, 29 

To find the horizontal parallax from 

the parallax at any altitude, 29 

Amount of parallax in the zenith and 

in the horizon, 30 

Effect of parallax upon the altitude of 

a body, 30 

Mode of determining the horizontal 

parallax of a body, 30 

Amount of the sun's hor. par 31 

Use of parallax, 31 

Refraction. Its effect upon the alti- 
tude of a body, 32 

[ts nature illustrated, 32 

[ts amount at different angles of eleva. 

tion, 32 

How the amount is ascertained, 33 

Sources of inaccuracy in estimating the 

refraction, 35 

Effect of refraction upon the sun and 

moon when near the horizon, 35 

Oval figure of these bodies explained,. 35 
Apparent enlargement of the sun and 

moon near the horizon, ! 

Twilight. Its cause explained, 37 

Length of twilight in different latitudes, 37 
How the atmosphere contributes to dif- 
fuse the sun's light, 37 

Chapter IV. TIME. 

Time defined, 38 

What period is a sidereal day, 38 

Uniformity of sidereal days, 38 

Solar time, how reckoned, 39 

Why solar days are longer than side- 
real, 

Apparent time defined, 39 

Mean time, 40 

An astronomical day, 40 

Equation of time defined, 40 

When do apparent time and mean 

time differ most? 40 

When do they come together ? 40 

Effect of a change in the place of the 

earth's perihelion, 40 

Causes of the inequality of the solar 

days, 41 

Explain the first cause, depending on 

the unequal velocities of the sun,.... 4l 



ANALYSIS. 



Vii 



Page. 
Explain the second cause, depending 

on the obliquity of the ecliptic, 42 

When does the sidereal day com- 
mence? 

The Calendar. Astronomical year de- 
fined, 45 

How the most ancient nations deter- 
mined the number of days in the year, 
Julius Caesar's reformation of the calen- 

dar explained, 45 

Errors of this calendar, 45 

Reformation by Pope Gregory, 46 

Rule for the Gregorian calendar, 46 

New style, when adopted in England, 46 
What nations still adhere to the old 

style? 

What number of days is now allowed 

between old and new style ? 47 

How the common year begins and ends, 

How leap year begins and ends, 47 

Does the confusion of different calen- 
dars affect astronomical observations ? 47 



Chapter V. ASTRONOMICAL INSTRUMENTS 
AND PROBLEMS FIGURE AND DENSITY OF 

THE EARTH. 

How the most ancient nations acquired 

their knowledge of Astronomy, 48 

Use of instruments in the Alexandrian 

School, 48 

Ditto, by Tycho Brahe, .' 48 

Ditto, by the Astronomers Royal, 48 

Space occupied by 1" on the limb of an 

instrument, 48 

Extent of actual divisions on the limb, 49 

Vernier, defined, 49 

Its use illustrated, 49 

Chief astronomical instruments enu- 
merated, 50 

Observations taken on the meridian. . . 50 

Reasons of this, 50 

Transit Instrument defined, 51 

Ditto described, 51 

Method of placing it in the meridian. . 51 

Line of collimation defined, 52 

System of wires in the focus, 

Its use for arcs of right ascension, 52 

Astronomical Clock, how regulated, 52 

What does it show ? 52 

How to test its accuracy, 53 

How corrected, 53 

Mural Circle, its object, 54 

Describe it, 54 

How the different parts contribute to 

theobject, 54 

Mural Quadrant, 55 

Use of the Mural Circle for arcs of de- 
clination, 56 

Altitude and Azimuth Instrument de- 
fined, 56 



Page. 

Its use, .. 56 

Describe it, ; 57 

Sextant described, 58 

44 How to measure the angular distance 

of the moon from the sun, 59 

How to take the altitude of a heavenly 
body 59 

45 Use of the artificial horizon, 59 

In what consists the peculiar value of 

theSextant? 60 

ASTRONOMICAL PROBLEMS. Given the 
the sun's right ascension and decli- 
nation, to find his longitude and the 

obliquity of the ecliptic, 61 

Napier's Rule of circular parts, 62 

46 Given the sun's declination to find his 

rising and setting at any place whose 
latitude is known, 63 

47 Given the latitude of a place and the 

declination of a heavenly body, to 
determine its altitude and azimuth 
when on the six o'clock hour circle, 64 
The latitudes and longitudes of two 
celestial objects being given, to find 

their distance apart, 65 

FIGURE AND DENSITY OF THE EARTH 
reason for ascertaining it with great 

precision, 66 

How found from the centrifugal force, 66 
From measuring an arc of the meridian, 67 
From observations with the pendulum, 68 

From the motions of the moon, 68 

Density of the earth compared with 

water, 68 

How ascertained by Dr. Maskelyne, . . 69 
Why an important element, 69 

Part II. -OF THE SOLAR SYSTEM 

Chapter 1. THE SUN SOLAR SPOTS Zo 
DIACAL LIGHT. 



Figure of the sun, '. 70 

Angle subtended by a line of 400 miles, 70 

Distance from the earth, 70 

Illustrated by motion on a railway car, 70 
52 Apparent diameter of the sun how 

found, 72 

How to find the linear diameter, 71 

How much larger is the sun than the 

earth, 71 

[ts density and mass compared with 

the earth's, 71 

Weight at the surface of the sun, 72 

Velocity of falling bodies at the sun ... 72 

SOLAR SPOTS. Their number, 72 

Size, 72 

Description, 72 

What region of the eun do they oc- 
cupy, 73 

Proof that they are on the sun, 73 



Vlll 



ANALYSIS. 



Page 
How we learn the revolution of the sun 

on his axis, 73 

Time of the revolution, 72 

Apparent paths of the spots, 

Inclination of the solar axis, 74 

Sun's Nodes when does the sun pass 

them? 75 

Cause of the solar spots, 76 

Faculffi, 76 

ZODIACAL LIGHT. Where seen, 76 

Its form, 76 

Aspects at different seasons, 

Its motions, 

Its nature,... . 77 



Page 

Product of the angle described in any 
given time by the square of the dis- 
tance, 8b 

74 Space described by the radius vector of 

the solar orbit in equal times, 88 

How to represent the sun's orbit by a 
diagram, 89 

Chapter III. UNIVERSAL GRAVITATION. 



Chapter II. APPARENT ANNUAL MOTION La 
OF THE SUN SEASONS FIGURE OF THE 
EARTH'S ORBIT. 



Apparent motion of the sun, 

How both the sun and earth are said to 

move from west to east, 79 

Nature and position of the sun's orbit, 

how determined, 

Changes in declination how found, 79 

Ditto, in right ascension, 

Inferences from a table of the sun's de- 
clinations, 80 

Ditto, of right ascensions, 81 

Path of the sun, how proved to be a 

great circle, 81 

Obliquity of the ecliptic, how found, 81 

How it varies, 81 

Great dimensions of the earth's orbit, 81 

Earth's daily motion in miles, 82 

Ditto, hourly ditto, 

Diurnal motion at the equator per 

hour, 

SEASONS. Causes of the change of sea- 



78 If 



79 In 



85111 



How each cause operates, 

Illustrated by a diagram, 

Change of seasons had the equator been 
perpendicular to the ecliptic, 84 

FIGURE OF THE EARTH'S ORBIT. Proof 
that the earth's orbit is not circular, 85 

Radius vector denned, 

Figure of the earth's orbit how ob- 
tained, : 

Relative distances of the earth from the 
sun, how found, 86 

Perihelion and Aphelion defined, 87 

Variations in the sun's apparent diame- 
ter, 87 

Angular velocities of the sun at the pe- 
rihelion and aphelion, 

Ratio of these velocities to the dis- 
tances, 87 

How to calculate the relative distances 
of the earth from the sun's daily mo- 
tions, 88 



Universal Gravitation defined, 90 

76 Why is it called attraction, 90 

77 History of its discovery, 90 

How was the gravitation of the moon 

to the earth first inferred ? 91 

ws of Gravitation. If a body re- 
volves about an immovable center 
of force, and is constantly attracted 

to it, how will it move ? 92 

a body describes a curve around a 
center towards which it tends by any 
force, how is its angular velocity re- 
lated to the distance, 93 

the same curve, the velocity at any 
point of the curve varies as what ? 93 

80 If equal areas be described about a cen- 
ter in equal times, to what must the 

force tend? 94 

How is the distance of any planet from 
the sun at any point in its orbit, to 
its distance from the* superior focus ? 94 
O ase of two bodies gravitating to the 
same center where one descends in a 
straight line, and the other revolves 
in a curve, 95 

82 Velocity of a body at any point when 

falling directly to the sun, 97 

82 Relation between the distances and pe- 
riodic times, 99 



82 Kepler's three great laws, 99 

82 MOTION IN AN ELLIPTICAL ORBIT, 100 



83 Idea of a projectile force, 100 

Nature of the impulse originally given 

to the earth, 100 

Two forces under which a body re- 
volves, 100 

ustrated by the motion of a cannon 

ball, 101 

86 Why a planet returns to the sun, 102 

Illustration by a suspended ball, 103 



Chapter IV. PRECESSION OF THE EQUI- 
NOXES NUTATION ABERRATION MEAN 
AND TRITE PLACES OF THE SUN. 

87 Precession of the Equinoxes defined, 104 

Why so called, 104 

Amount of Precession annually, 104 

Revolution of the equinoxes, 104 

Revolution of the pole of the equator 

around the pole of the ecliptic, 105 



ANALYSIS. 



IX 



Page 
Changes among the stars caused by 

precession, 

The present pole star not always such, 10' 
What will be the pole star 13,000 

years hence ? 105 

Cause of the precession of the equi- 
noxes, 105 

Explain how the cause operates, 10 

Proportionate effect of the sun and 

moon in producing precession, 107 

Tropical year defined, 107 

How much shorter than the sidereal 



year, 



107 



Use of the precession of the equinoxes 

in chronology, 107 

NUTATION, defined, 10 

Explain its operation, 108 

Cause of Nutation, 108 

ABERRATION, defined, 
Illustrated by a diagram, 
Amount of aberration, 

Effect on the places of the stars, 109 

MOTION OF THE APSIDES, the fact sta- 
ted, 109 

Direction of this motion, 11C 

Time of revolution of the line of Ap- 
sides, 110 

Present longitude of the perihelion, . . IK 

When was it nothing? 11C 

MEAN AND TRUE PLACES OF THE SUN, 111 

Mean Motion defined, Ill 

Illustrated by surveying a field, Ill 

Mean and true longitude distinguish- 



ed, 



111 



Equations defined, , 111 

Their object, Ill 

Mean and True anomaly defined, .... 112 

Equation of the Center, 112 

Explain from the figure, 112 

Chapter V. THE MOON LUNAR GEOGRA- 
PHY PHASES OF THE MOON HER REVO- 



Distance of the moon from the earth, 
Her mean horizontal parallax,... 



113 
113 



Her diameter, 113 How much more is the moon attract- 



Volume, density, and mass, 113 

Shines by reflected light, 113 

Appearance in the telescope, 113 

Terminator defined, 113 

Its appearance, 113 

Proofs of Valleys, 114 

Form of .these, 114 

Best time for observing the lunar 

mountains and valleys, 114 

Names of places on the moon double, 115 

Dusky regions how named, 115 

Point out remarkable places on the 

mapof the moon, 115 

Explain the method of estimating the 

height of lunar mountains, 115 



Page. 
Specify the heights of particular 

mountains, 117 

Volcanoes, proof of their existence, .... 117 

Has the moon an atmosphere ? 1 17 

Improbability of identifying artificial 

structures in the moon, 117 

PHASES OF THE MOON, their cause,.... 118 
Successive appearances of the moon 

from one new moon to another, .... 118 

Syzygies defined, 118 

Explain the phases of the moon from 

figure 46, , 119 

REVOLUTIONS OF THE MOON. Period 

of her revolutions about the earth, . 119 

Her* apparent orbit a great circle, 120 

A sidereal month defined..., 120 



A synodical do. 



120 



Length of each, 120 

Why the synodical is longer, 120 

How each is obtained, 120 

[nelination of the lunar orbit, 121 

Nodes defined, 121 

Why the moon sometimes runs high 

and sometimes low, 121 

Sarvest moon defined, < 122 

Ditto explained, 122 

Explain why the moon is nearer to us 
when on the meridian than when 

near the horizon, 122 

Time of the moon's revolution on its 

axis, 123 

low known, 123 

vibrations explained, 123 

Diurnal Libration, 124 

^ength of the Lunar days, .' 124 

Harth never seen on the opposite side 

of the moon, 124 

Appearances of the earth to a specta- 
tor on the moon, 124 

Why the earth would appear to re- 
main fixed, 125 

Ascending and descending nodes dis- 
tinguished, 125 

Whether the earth carries the moon 

around the sun, 126 



ed towards the sun than towards 

the earth, 126 

Vhen does the sun act as a disturbing 
force upon the moon ?...... 126 

Why does not the moon abandon the 
earth at the conjunction? 126 

?he moon's orbit concave towards the 

127 

How the elliptical motion of the moon . 
about the earth is to be conceived 
of, 127 

ilustrations, 127 

Chapter VI. LUNAR' IRREGULARITIES. 
Specify their general cause, 127 



ANALYSIS. 



Unequal action of the sun upon the 
earth and moon, 

Oblique action of earth and sun, 

Gravity of the moon towards the 
earth at the syzygies, 

Gravity at the quadratures, 

Explain the disturbances in the 
moon's motions from figure 48, 

Figure of the moon's orbit, 

How its figure is ascertained, 

Moon's greatest and least apparent di- 
ameters, 

Her greatest and least distances from 
the earth, 

Perigee and Apogee defined, 

Eccentricities of the solar and lunar 
orbits compared, , 

Moon's nodes, their change of place,. 

Rate of this change per annum, 

Period of their revolution, 

Irregular curve described by the 
moon, 

Cause of the retrograde motion of 
nodes, 

Explain from figure 50, 

Synodical revolution of the node de- 
fined, 

Its period, 

The Saros explained, 

The Metonic Cycle, 

Golden Number, 

Revolution of the line of apsides, 

Its period, 

How the places of the perigee may be 
found, 

Moon's anomaly defined, 

Cause of the revolution of the apsides, 

Amount of the equation of the Center, 

Evection defined, 

Its cause explained, 

Variation defined, 

Its cause, 

Annual Equation explained, 

How these irregularities were first 
discovered, 

How many equations are applied to 
the moon's motions? 

Method of proceeding in finding the 
moon's place, 

Successive degrees of accuracy at- 
tained, 

Periodic and secular irregularities dis- 
tinguished, 

Acceleration of the moon's mean mo- 
tion explained, 

Its consequences, 

Lunar inequalities of latitude and 

parallax, 

Chapter VII. ECLIPSES. 

Eclipse of the moon, when it happens, 



Page 

Eclipse of the sun, when it happens, . 
128 When only can each occur, 

128 Why an eclipse does not occur at 

every new and full moon, 

129 Why eclipses happen at two opposite 

129 months, 

Circumstances which affect the length 

130 of the earth's shadow, 

132 Semi-angle of the cone of the earth's 
132 shadow, to what equal, 

Length of the earth's shadow, 

Its breadth where it eclipses the 



132 
132 

132 Lunar ecliptic limit defined,. 

132 Solar, ditto 

Amount of the lunar ecliptic limit,.... 

133 Appulse defined,. 

133 Partial, total, central, eclipse, each 

133 defined,. 

133 Penumbra defined, 

Semi-angle of the moon's penumbra, 
133 to what equal,, 

Semi-angle of a section of the penum- 

133 bra where the moon crosses it, 

134 Moon's horizontal parallax increased 



135 
135 
135 
135 
136 
136 
136 

136 
136 
136 
137 
137 
138 
140 
140 
140 

141 
141 
141 
141 
141 

141 
142 

142 
143 



Why the moon is visible in a total 
eclipse, 

Calculation of eclipses, general mode 
of proceeding, 

To find the exact time of the begin- 
ning, end, duration, -and magnitude 
of a lunar eclipse, by figures 53, 54, 

Elements of an eclipse defined, 

Digits defined, 

How the shadow of the moon travels 
over the earth in a solar eclipse,.... 

Why the calculation of a solar eclipse 
is more complicated than a lunar,. 

Velocity of the moon's shadow, 

Different ways in which the shadow 
traverses the earth, according as 
the conjunction is near the node or 
near the limit, 

When do the greatest eclipses hap- 
pen? 

Case in which the moon's shadow 
nearly reaches the earth, 

How far may the shadow reach be- 
yond the center of the earth ? 

Greatest diameter of the moon's sha- 
dow where it traverses the earth,., 
reatest portion of the earth's surface 
ever covered by the moon's penum- 
bra, 

Moon's apparent diameter compared 
with the sun's, 

Annular eclipse, its cause, 

Direction in which the eclipse passes 
on the sun's disk, 



143 
143 

144 
144 
144 

145 
145 

146 
146 
146 
146 
147 

147 
147 

148 

148 
148 

148 
149 



150 
151 
153 

153 

154 
T54 



155 
155 
156 
157 
157 

157 

158 
158 

159 



ANALYSIS. 



XI 



Page. 

Greatest duration of total darkness,. . . 159 

Eclipses of the sun more frequent 
than of the moon, why? 159 

Lunar eclipses oftener visible, why ? 159 

Radiation of light in a total eclipse of 
the sun, 160 

Interesting phenomena of a total 
eclipse of the sun, 160 

Phenomena of the eclipse of 1806, de- 
scribed, 

When does the next total eclipse of 
the sun, visible . in the United 
States, occur? 161 



Chapter VIII. LONGITUDE. TIDES. 
Objects of the ancients in studying 

astronomy, 161 

Ditto of the moderns, 161 

LONGITUDE. How to find the differ, 
ence of longitude between two 

places, 161 

Method by the Chronometer explain. 

ed, 162 

How to set the chronometer to Green- 
wich time, 162 

Accuracy of come chronometers, 1 62 

Objections to them, 162 

Longitude by eclipses explained, 163 

Lunar method of finding the longi- 
tude, 163 

Circumstances which render this 

method somewhat difficult, 164 

Disadvantages of this method, 164 

Degree of accuracy attainable, 165 

TIDES. defined, 165 

High, Low, Spring, Neap, Flood, and 

Ebb Tide, severally denned, 165 

Similar tides on opposite sides of the 

earth, 165 

Interval between two successive high 

tides, 165 

Average height for the whole globe, 166 

Extreme height, 166 

Cause of the tides, 166 

Explain by figure 56, 166 

Tide-wave denned, 167 

Comparative effects of the sun and 

moon in raising the tide, 167 

Why the moon raises a higher tide 

than the sun, 167 

Springtides accounted for, 168 

Neap tides, ditto 168 

Power of the sun or moon to raise 
the tide, in what ratio to its /dis- 
tance, 168 

Influence of the declinations of the 

sun and moon on the tides, 169 

Explain from figures 57 and 58, 169 

Motion of the tide-wave not progres- 
sive, : 170 

Tides of rivers, narrow bays, how 
produced, 170 



Cotidal Lines defined, 170 

Derivative and Primitive tides distin- 
guished, 170 

Velocity of the tide-wave, circum- 
stances which affect it, 171 

Explain by figure 59, 171 

Examples of very high tides, 172 

Unit of altitude defined, 172 

Unit of altitude for different places, 172 
160 Tides on the coast of N. America, 

whence derived, 173 

Why no tides in lakes and seas, 173 

Intricacy of the problem of the tides, 173 

Atmospheric tide, 173 



Chapter IX. THE PLANETS INFERIOR 

PLANETS MERCURY AND VENUS. 
Signification of the term planet .... 174 

Planets known from antiquity 174 

Planets added in 1781 and in 1846 174 
Asteroids, their number and names 174 
Primary and Secondary Planets dis- 
tinguished 174 

Number of each 175 

Inclination of the planetary orbits 

to the ecliptio 175 

Inferior and Superior Planets dis- 
tinguished 175 

How the planets differ among them- 
selves 175 

Distances from the sun in miles .... 175 
Great dimensions of the planetary 

system 176 

Illustrated by the motion of a rail- 
way car 176 

Order by which the distances of the 

planets increase 176 

Bode's law of distances 177 

Mean distances, how determined... 177 

Diameters in miles 177 

Great diversity in respect to mag- 
nitude ' 178 

How the real diameters are found 

from the apparent 178 

Periodic Times in months and years 178 
Which of the planets move rapidly 

and which slowly 179 

INFERIOR PLANETS. Proximity to 

.the sun 179 

Illustration by Fig. 60 180 

Conjunction defined inferior and 

superior 180 

Sy nodical revolution defined 180 

How to find the synodical from the 

sidereal '. 181 

Motion of an inferior planet, when 

direct and when retrograde 181 

How these motions are affected by 

the earth's motions 182 

When the inferior planets are sta- 
tionary 182 



Xll 



ANALYSIS. 



Page. 

Elongation of the stationary points 

for Mercury and Venus 183 

Phases of the inferior planets 183 

Relative distances from the sun .... 183 

Eccentricity of their orbits 184 

Most favorable time for determining 

the sidereal period 184 

When is an inferior planet bright- 
est ? 185 

Diurnal revolutions of Mercury and 

Venus 185 

Venus as the morning and evening 

star 185 

Phenomena every eight years 186 

TRANSITS OF THE INFERIOR PLANETS 

DEFINED 186 

When they occur why not at ev- 
ery inferior conjunction 186 

Why those of Mercury in May and 
November 186 

Why those of Venus in June and 
December 186 

Intervals between the transits of 
Mercury 187 

Intervals between the transits of 
Venus 187 

How found 188 

Why so great an interest is attached 
to the transits of Venus 188 

Why the sun's horizontal parallax 
cannot be found like the moon's 189 

Why distant places of observation 
are taken 189 

Process for the sun's hor. par. ex- 
plained from Fig. 63 189 

Circumstances favorable to the ac- 
curacy of the result 190 

Sun's hor. par. in seconds 191 

To find the hor. par. of Venus and 
of Mars 191 

Atmosphere of Venus 191 

Satellites of Mercury and of Venus ? 191 

Chapter X. SUPERIOR PLANETS ASTER- 
OIDS MOTIONS OF THE PLANETS. 

Superior Planets, how distinguished 

from the Inferior 192 

MARS size distance from the sun 192 
Changes in apparent magnitude and 

brightness 193 

Phases of Mars, Fig. 64 193 

Telescopic appearances 1 94 

Satellite ? ellipticity 194 

To find the hor. par. of Mars 194 

JupfTER magnitude figure diur- 
nal revolution 195 

Inclination of the axis to the orbit, 

and change of seasons 195 

Telescopic appearances 195 

Belts described and explained 196 

Satellites how seen names 197 



Magnitudes distances periods . . . 

Orbits form inclination 

Eclipses then: various phenomena, 
Fig. 65 

Shadows cast by the satellites on 
the Primary 

Longitudes from the eclipses of Ju- 
piter's satellites 

Velocity of light, how discovered . . 

SATURN size ring telescopic 
view 

Ring described 

Dimensions of the system 

Position of the axis of rotation 

Rapid diurnal revolution 

Revolution of the ring around the 
sun 

Its changes and disappearances ex- 
plained 

Revolution of the ring in its own 
plane, how discovered 

Thickness of the ring new ring ... 

Satellites of Saturn number and 
names 

Eclipses 

URANUS its discovery 

Size periodic time inclination .... 

Satellites number peculiarities . . 

NEPTUNE distance diameter 
period 

History of its discovery 

Agreement of observation with 
theory 

Simultaneous discovery 

Results obtained by Walker 

ASTEROIDS history of the first four 

Distance from the sun size orbits 

Whole number 

PLANETARY MOTIONS two methods 
of studying them 

Appearances viewed from the sun 

Motions of Mercury explained 

Three things to be regarded in the 
planetary orbits 

Why diagrams and orreries repre- 
sent them erroneously 

Apparent motions of the planets... 

Two causes make them unlike the 
real 

Apparent motions illustrated by 
Fig. 69 

Apparent motions of the Superior 
Planets 

Illustrated by Fig. 70 



ANALYSIS. 



X1U 



Notions of Ptolemy and Hipparchus 

Kepler Investigation of the mo- 
tions of Mars 

Discovery of the first law the 
second the third 

Modification of the third law 

ELEMENTS OF THE PLANETARY ORB- 
ITS enumerated 

Why not found like the lunar and 
solar orbits 

First steps of the process for finding 
the elements 

To convert geocentric longitudes 
and latitudes into heliocentric, 
Fig. 71 

To determine the position of the 
nodes 

To determine the inclination 

To find the periodic time 

The position of a planet which is 
most favorable for finding the 
elements 

Exemplified in finding the period- 
ic time of Saturn 

To determine the distance from the 
sun 

How the mean distance is found ... 

How the distance at any point in 
the orbit 

Method for the Inferior Planets.... 

Method for the Superior, Fig. 73 ... 

To determine the place of the peri- 
helion 

To determine the epoch of posing 
the perihelion 

To find the eccentricity 

QUANTITY OF MATTER IN THE SUN 
AND PLANETS 

How found in terms of the distances 
and periodic times 

How found by the spaces fallen 
through, Fig. 75 

How found in planets which have 
no satellites 

Densities, how found 

Specific gravities of the sun and 
planets respectively 

Comparative densities 



Page 

224 

225 

225 
227 

227 
227 
228 

228 

220 
23C 
230 

231 
231 

232 

232 

232 
232 
232 

233 

235 
235 

236 
236 
237 



238 
238 

239 
239 

Chapter XII. PERTURBATIONS OF THE 
PLANETS STABILITY OF THE SYSTEM 
NUMERICAL RELATIONS PROBLEMS. 

PERTURBATIONS Numerous causes 

Case where the only bodies are a 
central and a revolving body 240 

How these irregularities have been 
discovered 241 

Periodical and secular perturba- 
tions distinguished 241 

Example Changes of eccentricity 
of the earth's orbit 242 



Whether the perturbations accumu- 
late indefinitely 242 

Stability of the system how main- 
tained 242 

Nature of the evidence to prove 

the stability ^ 243 

Invariability of the grand axes P 243 

Limits to the variation of the ec- 
centricity 243 

Also to that of the inclination 244 

What kind of perturbations are cu- 
mulative and what are oscilla- 
tory 244 

Conditions essential to thjs stabil- 
ity .. 244 

Long inequality of Jupiter and 

Saturn 244 

Also of the earth and Venus 245 

NUMERICAL RELATIONS OF THE PLAN- 
ETARY SYSTEM 245 

Change of velocity necessary on in- 
creasing the mass 245 

Also on increasing the distance .... 245 
Members of the solar system, how 

adjusted * 246 

Relation between the rate of mo- 
tion, distance, periodic time, and 

force of gravity 246 

Demonstration of the rules 246 

The rules stated 247 

Given, the rate of motion, to find 

the other terms 247 

Given, the distance 247 

Given, the periodic time 247 

Given, the force of gravitation 248 

Required, the rate of motion, dis- 
tance, period, and force of gravi- 
tation respectively 248 

Problems 249 

Chapter XIII. COMETS METEORIC 
SHOWERS. 



COMETS their several parts 252 

Number belonging to the System.. 253 

The six most remarkable 253 

Variations in magnitude and bright- 
ness 254 

To what owing 255 

Periods of revolution 255 

Distances from the sun 256 

Figure of the orbit of Halley's com- 
et 256 

240 Source of the light 256 

Direction of the trains 257 

Quantity of matter in comets 257 

How the orbit of a comet may be 

changed 258 

Example in the comet of 1770 258 

DRBITS AND MOTIONS OK COMETS 260 

How they differ from those of plan- 
ets , 260 



XIV 



ANALYSIS. 



Page. 

Elements enumerated 260 

Their investigation, why difficult... 260 
How the return of a comet is pre- 
dicted 262 

Exemplified in Halley's comet 263 

Its ra|urn in 1759 and 1835 264 

Whyiui astronomical event of great 

interest 264 

Encke 's comet its period 265 

Question of a resisting medium 265 

Comet of 1843 its remarkable pe- 
culiarities 266 

Physical nature of comets 267 

Possibility of. their striking the 

earth 268 

METEORIC SHOWERS great shower 

of Nov. 1833 270 

Point of apparent radiation 270 

Extent and duration 270 

Periods of its recurrence 271 

Why an astronomical or cosmical 

phenomenon 271 

Of the periods of meteoric showers 271 
Conclusions respecting the meteors, 
as to their origin, nature, veloci- 
ty, size, light, and heat 271 

Reasons for these conclusions 272 

Part III OP THE FIXED STARS 
AND SYSTEM OF THE WORLD. 

Chapter I. FIXED STARS CONSTELLA- 
TIONS. 

Why called fixed stars 274 

Classification 274 

Number in each class 274 

Antiquity of the constellations 275 

Their names how individual stars 

are denoted 275 

Catalogues of the stars 275 

Number in the catalogue of Hip- 

parchus 

Number inLalande's 276 

Utility of learning the constella- 
tions 276 

Constellations of the Zodiac Aries, 

Taurus 277 

Seven stars in Pleiades 278 

Gemini, Cancer 278 

Praesepe, or the Bee-hive 279 

Leo, Virgo, Libra 279 

Scorpio, Sagittarius, Capricornus, 

Aquarius, Pisces 280 

Northern Constellations 281 

Ursa Minor, Ursa Major 281 

Draco 282 

Cepheus, Cassiopeia, Camelopard, 

Andromeda 283 

Perseus, Auriga, Leo Minor, Canes 

Venatici, Coma Berenices, Bootes, 284 



Page. 
Corona Borealis, Hercules, Lyra, 

Gygnus 285 

Vulpecula, Aquila, Antinous, Del- 

phinus, Pegasus, Ophiuchus 286 

Southern Constellations 286 

Orion, Lepus, Cam's Major 287 

Canis Minor, Menoceros, Hydra 288 

Lesson for the middle of September 288 

Lesson for the middle of December. 289 

Lesson for the middle of March .... 290 

Lesson for the middle of June 290 

Chapter II. DOUBLE STARS TEMPORARY 
STARS VARIABLE STARS CLUSTERS 
AND NEBULAE. 

Use of great telescopes in studying 

the stars 291 

Herschel's 40-feet telescope 291 

Rosse telescope 292 

Pulkova and Cambridge telescopes 292 

DOUBLE STARS defined 293 

By whom discovered 293 

Examples number 293 

When merely optically double 294 

When physically double 294 

System of double, triple, and mul- 
tiple stars 294 

Colors of the components 294 

TEMPORARY STARS defined 294 

Examples 295 

VARIABLE STARS defined 295 

Examples. 

Evidence of activity among the 

stars....* 295 

CLUSTERS examples 296 

NEBULAE defined 297 

Examples nebulas of Andromeda . 297 

Nebula of Hercules 297 

Magellanic clouds 297 

Nebula of Orion 298 

Use of great telescopes for these 

objects 298 

ingular forms of nebulae 298 

Resolvable and irresolvable dis- 
tinguished 298 

Signs of beauty and symmetry 

among the nebulae 299 

Nebulous Stars defined 299 

Annular Nebulae defined 299 

Example in Lyra 299 

Planetary Nebulae 299 

Resemblance to planets great ex- 
tent 299 

Example in Andromeda 300 

Milky Way cause of its peculiar 

light 300 

Number of its component stars .... 300 



276 Singul 



Binary Stars defined 



ANALYSIS. 



Number of these 

Periodic times examples 

Law of gravitation among the stars 

Proper Motions of the stars 

Result on comparing the places of 
certain stars in ancient and mod- 
ern catalogues 

Motion of the solar system in space 

Point toward which it is moving . . . 

Rate of motion per annum 

Examples of great annual proper 
motions 

DISTANCES OF THE STARS how found 

What is the base line for parallax . 

Why it was supposed impossible to 
determine a parallax of less than 
1' 

Distance implied by a par. of 1"... 

Bessel's determination of the par. 
of 61 Cygni 

His method of investigation 

Distance measured by the progress 
of light and by a railway car, 
respectively 

Actual period of revolution of the 
components of 61 Cygni 

Space described by the star annu- 
ally 

Reliance to be placed on Bessel's 
determination 

Nature of the Stars 

Size of Sirius compared with the 
sun 

Proof that the fixed stars are 
suns 

End for which they were made 

Arguments for a plurality of worlds 



801 Chapter IV. SYSTEM OF THE WOKLD. 

801 Page. 

802 System of the world defined 310 

802 Complex character of early sys- 

tems 310 

Things known to Pythagoras 310 

303 His visionary notions 811 

803 Rejection of his system 311 

804 Crystalline spheres of Eudoxus .... 312 
804 How the two motions were ac- 
counted for -312 

305 Hipparchus truths discovered by 

305 him 312 

306 Almagest of Ptolemy 312 

Ptolemaic System explained 813 

Illustrated by Fig. 81 314 

306 Defects of this system 815 

306 Objections to it 315 

Tychonic 8y* ton explained 815 

807 Objections to it 316 

307 Copernican System explained 816 

Arguments on which it rests 316 

Proofs that the planets revolve 

307 about the sun 317 

Proofs of systems among the stars. 317 

307 Exemplified in the Pleiades, Nebu- 

la of Hercules, Binary Stars, and 

807 Nebulee 318 

Uniformity Q^ plan in natural struc- 

308 tures 318 

308 Ascending orders of systems de- 

308 scribed 318 

308 Supposed centre of the universe ... 318 
Central sun where placed... 319 

309 Reasons for believing that all the 
809 heavenly bodies are united in one 

309 grand system 319 



[O 3 Diagrams for public recitations. 



As many of the figures of this work are too complicated to be 
drawn on the black-board at each recitation, we have found it 
very convenient to provide a set of permanent cards of paste- 
board, on which the diagrams are inscribed on so large a scale, as 
to be distinctly visible in all parts of the lecture room. The let- 
ters may be either made with a pen, or better procured of the 
printer, and pasted on. 

The cards are made by the bookbinder, and consist of a thick 
paper board about 18 by 14 inches, on each side of which a white 
sheet is pasted, with a neat finish around the edges. A loop at- 
tached to the top is convenient for hanging the card on a nail. 

Cards of this description, containing diagrams for the whole 
course of mathematical and philosophical recitations, have been 
provided in Yale College, and are found a valuable part of our ap- 
paratus of instruction. 



INTRODUCTION TO ASTRONOMY. 



PRELIMINARY OBSERV>t,toftS. 



1. ASTRONOMY is that science which treats of the heavenly Indies. 

More particularly, its object is to teach what is known respect- 
ing the Sun, Moon, Planets, Comets, and Fixed Stars ; and also to 
explain the methods by which this knowledge is acquired. Astron- 
omy is sometimes divided into Descriptive, Physical, and Practi- 
cal. Descriptive Astronomy respects facts ; Physical Astronomy 
causes; Practical Astronomy, the means of investigating the facts, 
whether by instruments, or by calculation. It is the province of 
Descriptive Astronomy to observe, classify, and record, all the 
phenomena of the heavenly bodies, whether pertaining to those 
bodies individually, or resulting from their motions and mutual 
relations. It is the part of Physical Astronomy to explain the 
causes of these phenomena, by investigating and applying the 
general laws on which they depend ; especially by tracing out all 
the consequences of the law of universal gravitation. Practical 
Astronomy lends its aid to both the other departments. 

2. Astronomy is the most ancient of all the sciences. At a 
period of very high antiquity, it was cultivated in Egypt, in Chal- 
dea, in China, and in India. Such knowledge of the heavenly 
bodies as could be acquired by close and long continued observa- 
tion, without the aid of instruments, was diligently amassed ; and 
tables of the celestial motions were constructed, which could be 
used in predicting eclipses, and other astronomical phenomena. 

About 500 years before the Christian era, Pythagoras, of 
Greece* taught astronomy at the celebrated school at Crotona, and 
exhibited more correct views of the nature of the celestial mo- 
tions, than were entertained by any other astronomer of the an- 
cient world. His views, however, were not generally adopted. 

1 



IO 3 Diagrams for public recitations. 



As many of the figures of this work are too complicated to be 
drawn on the black-board at each recitation, we have found it 
very convenient to provide a set of permanent cards of paste- 
board, on which the diagrams are inscribed on so large a scale, as 
to be distinctly visible in all parts of the lecture room. The let- 
ters may be either made with a pen, or better procured of the 
printer, and pasted on. 

The cards are made by the bookbinder, and consist of a thick 
paper board about 18 by 14 inches, on each side of which a white 
sheet is pasted, with a neat finish around the edges. A loop at- 
tached to the top is convenient for hanging the card on a nail. 

Cards of this description, containing diagrams for the whole 
course of mathematical and philosophical recitations, have been 
provided in Yale College, and are found a valuable part of our ap- 
paratus of instruction. 



INTRODUCTION TO ASTRONOMY. 



PRELIMINARY 

1. ASTRONOMY is that science which treats of the heavenly bodies. 

More particularly, its object is to teach what is known respect- 
ing the Sun, Moon, Planets, Comets, and Fixed Stars ; and also to 
explain the methods by which this knowledge is acquired. Astron- 
omy is sometimes divided into Descriptive, Physical, and Practi- 
cal. Descriptive Astronomy respects facts ; Physical Astronomy 
causes; Practical Astronomy, the means of investigating thefacts t 
whether by instruments, or by calculation. It is the province of 
Descriptive Astronomy to observe, classify, and record, all the 
phenomena of the heavenly bodies, whether pertaining to those 
bodies individually, or resulting from their motions and mutual 
relations. It is the part of Physical Astronomy to explain the 
causes of these phenomena, by investigating and applying the 
general laws on which they depend ; especially by tracing out all 
the consequences of the law of universal gravitation. Practical 
Astronomy lends its aid to both the other departments. 

2. Astronomy is the most ancient of all the sciences. At a 
period of very high antiquity, it was cultivated in Egypt, in Chal- 
dea, in China, and in India. Such knowledge of the heavenly 
bodies as could be acquired by close and long continued observa- 
tion, without the aid of instruments, was diligently amassed ; and 
tables of the celestial motions were constructed, which could be 
used in predicting eclipses, and other astronomical phenomena. 

About 500 years before the Christian era, Pythagoras, of 
Greece* taught astronomy at the celebrated school at Crotona, and 
exhibited more correct views of the nature of the celestial mo- 
tions, than were entertained by any other astronomer of the an- 
rient world. His views, however, were not generally adopted. 

1 



2 PRELIMINARY OBSERVATIONS. 

but lay neglected for nearly 2000 years, when they were revived 
and established by Copernicus and Galileo. The most celebrated 
astronomical school of antiquity was at Alexandria, in Egypt, 
which was established and sustained by the Ptolemies, (Egyptian 
princes,) about 300 years before the Christian era. The employ- 
ment of instruments for; measuring angles, and the introduction of 
trigononietrical'caliulsiions to aid the naked powers of observa- 
tion, gave to the Alexandrian astronomers great advantages over 
all their predecessors. The most able astronomer of the Alexan- 
drian school was Hipparchus, who was distinguished above all the 
ancients for the accuracy of his astronomical measurements and 
determinations. The knowledge of astronomy possessed by the 
Alexandrian school, and recorded in the Almagest, or great work 
of Ptolemy, constituted the chief of what was known of our 
science during the middle ages, until the fifteenth and sixteenth 
centuries, when the labors of Copernicus of Prussia, Tycho Brake 
of Denmark,* Kepler of Germany, and Galileo of Italy, laid the 
solid foundations of modern astronomy. Copernicus expounded 
the true theory of the celestial motions ; Tycho Brahe carried 
the use of instruments and the art of astronomical observation to 
a far higher degree of accuracy than had ever been done before ; 
Kepler discovered the great laws of the planetary motions ; and 
Galileo, having first enjoyed the aid of the telescope, made innu- 
merable discoveries in the solar system. Near the beginning of 
the eighteenth century, Sir Isaac Newton discovered, in the law 
of universal gravitation, the great principle that governs the ce- 
lestial motions ; and recently, La Place has more fully completed 
what Newton began, having followed out all the consequences of 
the law of universal gravitation, in his great work, the Mecan- 
ique Celeste. 

3. Among the ancients, astronomy was studied chiefly as sub- 
sidiary to astrology. ASTROLOGY was the art of divining future 
events by the stars. It was of two kinds, natural and judicial. 
Natural Astrology, aimed at predicting remarkable occurrences in 
the natural world, as earthquakes, volcanoes, tempests, and pesti- 
lential diseases. Judicial Astrology, aimed at foretelling the fates 
of individuals, or of empires. 



PRELIMINARY OBSERVATIONS* 3 

4. Astronomers of every age, have been distinguished for their 
persevering industry, and their great love of accuracy. They 
have uniformly aspired to an exactness in their inquiries, far be- 
yond what is aimed at in most geographical investigations, satis- 
fied with nothing short of numerical accuracy, wherever this is 
attainable ; and years of toilsome observation, or laborious calcu- 
lation, have been spent with the hope of attaining a few seconds 
nearer to the truth. Moreover, a severe but delightful labor is 
imposed on all who would arrive at a clear and satisfactory knowl- 
edge of the subject of astronomy. Diagrams, artificial globes, 
orreries, and familiar comparisons and illustrations, proposed by 
the author or the instructor, may afford essential aid to the learner, 
but nothing can convey to him a perfect comprehension of the 
celestial motions, without much diligent study and reflection. 

5. In expounding the doctrines of astronomy, we do not, as in 
geometry, claim that every thing shall be proved a soon as as- 
serted. We may first put the learner in possession of the leading 
facts of the science, and afterwards explain to him the methods 
by which those facts were discovered, and by which they may 
be verified ; we may assume the principles of the true system of 
the world, and employ those principles in the explanation of many 
subordinate phenomena, while we reserve the discussion of the 
merits of the system itself, until the learner is extensively ac- 
quainted with astronomical facts, and therefore better able to ap- 
preciate the evidence by which the system is established. 

6. The Copernican System is that which is held to be the true 
system of the world. It maintains (1,) That the apparent diur- 
nal revolution of the heavenly bodies, from east to west, is owing 
to the real revolution of the earth on its own axis from west to 
east, in the same time ; and (2,) That the sun is the center around 
which the earth and planets all revolve from west to east, con- 
trary to the opinion that the earth is the center of motion of the 
sun and planets. 

7. We shall treat, first, of the Earth in its astronomical rela- 
tions ; secondly, of the' Solar System; and, thirdly, of the Fixed 
Stars. 



PART I. OF THE EARTH. 



CHAPTER I. 

OF THE FIGURE AND DIMENSIONS OF THE EARTH, AND THE DOCTRINE 



OF THE SPHERE. 



8. The figure of the earth is nearly globular. This fact is 
known, first, by the circular form of its shadow cast upon the 
moon in a lunar eclipse ; secondly, from analogy, each of the 
other planets being seen to be spherical ; thirdly, by our seeing 
the tops of distant objects while the other parts are invisible, as 
the topmast of a ship, while either leaving or approaching the 
shore, or the lantern of a light-house, which, when first descried 
at a distance at sea, appears to glimmer upon the very surface of 
the water ; fourthly, by the depression or dip of the horizon when 
the spectator is on an eminence ; and, finally, by actual observa- 
tions and measurements, made for the express purpose of ascer- 
taining the figure of the earth, by means of which astronomers are 
enabled to compute the distances from 

the center of the earth of various places Fl * L 

on its surface, which distances are found 
to be nearly equal. 

9. The Dip of the Horizon, is the ap- 
parent angular depression of the hori- 
zon, to a spectator elevated above the 
general level of the earth. The eye 
thus situated, takes in more than a ce- 
lestial hemisphere, the excess being the 
measure of the dip. * 

Thus, in Fig. 1, let AO represent the 




FIGURE AND DIMENSIONS. 5 

height of a mountain, ZO the direction of the plumb line, HOR a 
line touching the earth at the point O, and at right angles to the 
plumb line, C the center of the earth, DAE the portion of the 
earth's surface seen from O; OD, OE, lines drawn from the 
place of the spectator to the most distant parts of the horizon, 
and CD a radius of the earth. The dip of the horizon is the an- 
gle HOD or ROE. Now the angle made between the direction 
of the plumb line and that of the extreme line of the horizon or 
the surface of the sea, namely, the angle ZOD, can be easily 
measured ; and subtracting the right angle ZOH from ZOD, the 
remainder is the dip of the horizon, from which the length of the 
line OD may be calculated, (see Art. 10,) the height of the spec- 
tator, that is, the line OA, being known. This length, to whatever 
point of the horizon the line is drawn, is always found to be the 
same ; and hence it is inferred, that the boundary which limits 
the view on all sides, is a circle. Moreover, at whatever elevation 
the dip of the horizon is taken, in any part of the earth, the 
space seen by the spectator is always circular. Hence the sur 
face of the earth is spherical. 

10. The earth being a sphere, the dip of the horizon HOD= 
OCD. Therefore, to find the dip of the horizon corresponding 
to any given height' AO* (the diameter of the earth being known,) 
we have in the triangle OCD, the right angle at D, and the two 
sides CD, CO, to find the angle OCD. Therefore, 

CO : rad. : : CD : cos. OCD. Learning the dip corresponding 
to different altitudes, by giving to the line AO different values, 
we may arrange the results in a table. 

* The learner will remark that the line AO, as drawn in the figure, is much larger 
in proportion to CA than is actually the case, and that the angle HOD is much too 
great for the reality. Such disproportions are very frequent in astronomical diagrams, 
especially when some of the parts are exceedingly small compared with others ; and 
hence the diagrams employed in astronomy are not to be regarded as true pictures of 
the magnitudes concerned, but merely as representing their abstract geometrical re- 
lations. 



THE EARTH. 



Table showing the Dip of the Horizon at different elevations, from 
I foot to 100 feet.* 



\ Feet. 


/ // 


Feet. 


/ // 


Feet. 


/ // 


1 


0.59 


13 


3.33 


26 


5.01 


2 


1.24 


14 


3.41 


28 


5.13 


3 


1.42 


15 


3.49 


30 


5.23 


4 


1.58 


16 


3.56 


35 


5.49 


5 


2.12 


17 


4.03 


40 


6.14 


6 


2.25 


18 


4.11 


45 


6.36 


7 


2.36 


19 


4.17 


50 


6.58 


8 


2.47 


20 


4.24 


60 


7.37 


9 


2.57 


21 


4.31 


70 


8.14 


10 


3.07 


22 


4.37 


80 


8.48 


11 


3.16 


23 


4.43 


90 


9.20 


12 


3.25 


24 


4.49 


100 


9.51 



Such a table is of use in estimating the altitude of a body 
above the horizon, when the instrument (as usually happens) is 
more or less elevated above the general level of the earth. For 
if it is a star whose altitude above the horizon is required, the 
instrument being situated at O, (Fig. 1,) the inquiry is how far 
the star is elevated above the level HOR, but the angle taken is 
that above the visible horizon OD. The dip, therefore, or the 
angle HOD, corresponding to the height of the point O, must be 
subtracted, to obtain the true altitude. On the Peak of Tene- 
rifFe, a mountain 13,000 feet high, Humboldt observed the surface 
of the sea to be depressed on all sides nearly 2 degrees. The 
sun arose to him 12 minutes sooner than to an inhabitant of the 
plain ; and from the plain, the top of the mountain appeared en- 
lightened 12 minutes before the rising or after the setting of 
the sun. 

11. The foregoing considerations show that the form of the 
earth is spherical ; but more exact determinations prove, that the 
earth, though nearly globular, is not exactly so : its diameter from 
the north to the south pole is about 26 miles less than through 
the equator, giving to the earth the form of an oblate spheroid,f 

* This table includes the allowance for refraction. 

t An oblate spheroid is the solid described by the revolution of an ellipse about its 
shorter axis. 



FIGURE AND DIMENSIONS. 7 

x 

or a flattened sphere resembling an orange. We shall reserve the 
explanations of the methods by which this fact is established, 
until the learner is better prepared than at present to understand 
them. 

12. The mean or average diameter of the earth, is 7912.4 miles, 
a measure which the learner should fix in his memory as a stand- 
ard of comparison in astronomy, and of which he should endeavor 
to form the most adequate conception in his power. The circum- 
ference of the earth is about 25,000 miles (24857.5).* Although 
the surface of the earth is uneven, sometimes rising in high moun- 
tains, and sometimes descending in deep valleys, yet these eleva- 
tions and depressions are so small in comparison with the immense 
volume of the globe, as hardly to occasion any sensible deviation 
from a surface uniformly curvilinear. The irregularities of the 
earth's surface in this- view, are no greater than the rough points 
on the rind of an orange, which do not perceptibly interrupt its 
continuity ; for the highest mountain on the globe is only about 
five miles above the general level ; and the deepest mine hitherto 

opened is only about half a mile.f Now - = or about 



one sixteen hundredth part of the whole diameter, an inequality 
which, in an artificial globe of eighteen inches diameter, amounts 
to only the eighty-eighth part of an inch. 

I 
13. The diameter of the earth, con- 

sidered as a perfect sphere, may be de- 
termined by means of observations on 
a mountain of known elevation, seen 
in the horizon from the sea. Let BD 
(Fig. 2,) be a mountain of known 
height a, whose top is seen in the hori- 
zon by a spectator at A, b miles from it. 
Let x denote the radius of the earth. 
Then a? + & = (x+a)* = or 2 + 2ax + a?. 

* It will geerally be sufficient to treasure up in the memory the round number, 
but sometimes, in astronomical calculations, the more exact number may be required, 
and it is therefore inserted. 

t Sir John Herschel. 




8 THE EARTH. 



_ tt 

Hence, 2ax=tfcP, and a?= - . For example, suppose the 



height of the mountain is just one mile ; then it will be found, 
by observation, to be visible on the horizon at the distance of 

89 miles=&. Hence, ^ZL^-^^^i-T^lzl^gGO^radius 

2ci A 2 

of the earth, and 7920 =the earth's diameter. 

14. Another method, and the most anciejit, is to ascertain the 
distance on the surface of the earth, corresponding to a degree of 
latitude. Let us select two convenient places, one lying directly 
north of the other, whose difference of latitude is known. Sup- 
pose this difference to be 1 30', and the distance between the 
two places, as measured by a chain, to be 104 miles. Then, 
since there are 360 degrees of latitude in the entire circumference, 

24960 

1 30' : 104 : : 360 : 24960. And - - =7944. 

3.1416 

The foregoing approximations are sufficient to show that the 
earth is about 8,000 miles in diameter. 

15. The greatest difficulty in the way of acquiring correct 
views in astronomy, arises from the erroneous notions that pre- 
occupy the mind. To divest himself of these, the learner should 
conceive of the earth as a huge globe occupying a small portion 

Fig. 3. 




DOCTRINE OF THE SPHERE. 9 

of space, and encircled on all sides with the starry sphere. He 
should free his mind from its habitual proneness to consider one 
part of space as naturally up and another down, and view him- 
self as subject to a force which binds him to the earth as truly as 
though he were fastened to it by some invisible cords or wires, as 
the needle attaches itself to all sides of a spherical loadstone. He 
should dwell on this point until it appears to him as truly up in 
the direction of BB , CC , DD , (Fig. 3,) when he is at B, C, and 
D, respectively, as in the direction of AA when he is at A. 

DOCTRINE OF THE SPHERE. 

16. The definitions of the different lines, points, and circles, 
which are used in astronomy, and the propositions founded upon 
them, compose the Doctrine of the Sphere.* 

17. A section of a sphere by a plane cutting it in any manner, 
is a circle. Great circles are those which pass through the center 
of the sphere, and divide it into two equal hemispheres : Small 
circles, are such as do not pass through the center, but divide the 
sphere into two unequal parts. Every circle, whether great or 
small, is divided into 360 equal parts called degrees. A degree, 
therefore, is not any fixed or definite quantity, but only a certain 
aliquot part of any circle. 

18. The Axis of a circle, is a straight line passing through its 
center at right angles to its plane. 

19. The Pole of a great circle, is the point on the sphere where 
its axis cuts through the sphere. Every great circle has two 
poles, each of which is every where 90 from the great circle. 
For, the measure of an angle at the center of a sphere, is the 
arc of a great circle intercepted between the two lines that con- 
tain the angle ; and, since the angle made by the axis and any 
radius of the circle is a right angle, consequently its measure on 
the sphere, namely, the distance from the pole to the circumfer- 

* It is presumed that many of those who read this work, will Have studied Spherical 
Geometry ; but it is so important to the student of astronomy to have a clear idea of 
the circles of the sphere, that it is thought best to introduce them here. 

2 



10 THE EARTH. 

ence of the circle, must be 90. If two great circles cut each 
other at right angles, the poles of each circle lie in the circum- 
ference of the other circle. For each circle passes through the 
axis of the other. 

" 20. All great circles of the sphere cut each other in two points 
diametrically opposite, and consequently, their points of section 
are 180 apart. For the line of common section, is a diameter 
in both circles, and therefore bisects both. 

21. A great circle which passes through the pole of another 
great circle, cuts the latter at right angles. For, since it passes 
through the pole and the center of the circle, it must pass through 
the axis ; which being at right angles to the plane of the circle, 
every plane which passes through it is at right angles to the same 
plane. 

The great circle which passes through the pole of another great 
circle and is at right angles to it, is called a secondary to that circle. 

22. The angle made by two great circles on the surface of the 
sphere, is measured by the arc of another great circle, of which 
the angular point is the pole, being the arc of that great circle 
intercepted between those two circles. For this arc is the meas- 
ure of the angle formed at the center of the sphere by two radii, 
drawn at right angles to the line of common section of the two 
circles, one in one plane and' the other in the other, which angle 
is therefore that of the inclination of those planes. 

23. In order to fix the position of any plane, either on the sur- 
face of the earth or in the heavens, both the earth and the heav- 
ens are conceived to be divided into separate portions by circles, 
which are imagined to cut through them in various ways. The 
earth thus intersected is called the terrestrial, and the heavens the 
celestial sphere. The learner will remark, that these circles have 
no existence in nature, but are mere landmarks, artificially con- 
trived for convenience of reference. On account of the immense 
distance of the heavenly bodies, they appear to us, wherever we 
are placed, to be fixed in the same concave surface, or celestial 



DOCTRINE OF THE SPHERE. 11 

vault. The great circles of the globe, extended every way to 
meet the concave surface of the heavens, become circles of the 
celestial sphere. 

24. The Horizon is the great circle which divides the earth 
into upper and lower hemispheres, and separates the visible heav- 
ens from the invisible. This is the rational horizon. The sen- 
sible horizon, is a circle touching the earth at the place of the 
spectator, and is bounded by the line in which the earth and skies 
seem to meet. The sensible horizon is parallel to the rational, 
but is distant from it by the semi-diameter of the earth, or nearly 
4,000 miles. Still, so vast is the distance of the starry sphere, 
that both these planes appear to cut that sphere in the same line ; 
so that we see the same hemisphere of stars that we should see if 
the upper half of the earth were removed, and we stood on the 
rational horizon. 

25. The poles of the horizon are the zenith and nadir. The 
Zenith is the point directly over our head, and the Nadir that di- 
rectly under our feet. The plumb line is in the axis of the hori- 
zon, and consequently directed towards its poles. 

Every place on the surface of the earth has its own horizon , 
and the traveller has a new horizon at every step, always extend- 
ing 90 degrees from his zenith in all directions. 

26. Vertical circles are those whicfy pass through the poles of 
the horizon, perpendicular to it. 

The Meridian is that vertical circle which passes through the 
north and south points. 

The Prime Vertical, is that vertical circle which passes through 
the east and west points. 

27. As in geometry, we determine the position of any point by 
means of rectangular coordinates, or perpendiculars drawn from 
the point to planes at right angles to each other, so in astron- 
omy we ascertain the place of a body, as a fixed star, by taking 
its angular distance from two great circles, one of which is per- 
pendicular to the other. Thus the horizon and the meridian, or the 



12 THE EARTH. , 

horizon and the prime vertical, are coordinate circles used for such 
measurements. 

The Altitude of a body, is its elevation above the horizon meas- 
ured on a vertical circle. 

The Azimuth of a body, is its distance measured on the hori- 
zon from the meridian to a vertical circle passing through the body. 

The Amplitude of a body, is its distance on the horizon, from 
the prime vertical, to a vertical circle passing through the body. 

Azimuth is reckoned 90 from either the north or south point ; 
and amplitude 90 from either the east or west point. Azimuth 
and amplitude are mutually complements of each other. When a 
point is on the horizon, it is only necessary to count the number 
of degrees of the horizon between that point and the meridian, 
in order to find its azimuth ; but if the point is above the horizon, 
then its azimuth is estimated by passing a vertical circle through 
it, and reckoning the azimuth from the point where this circle cuts 
the horizon. 

The Zenith Distance of a body is measured on a vertical cir- 
cle, passing through that body. It is the complement of the alti- 
tude. 

28. The Axis of the Earth is the diameter, on which the earth 
is conceived to turn in its diurnal revolution. The same line con- 
tinued until it meets the starry concave, constitutes the axis of the 
celestial sphere. 

The Poles of the Earth #re the extremities of the earth's axis : 
the Poles of the Heavens, the extremities of the celestial axis. 

29. The Equator is a great circle cutting the axis of the earth 
at right angles. Hence the axis of the earth is the axis of the 
equator, and its poles are the poles of the equator. The intersec- 
tion of the plane of the equator with the surface of the earth, 
constitutes the terrestrial, and with the concave sphere of the 
heavens, the celestial equator. The latter, by way of distinction, 
is sometimes denominated the equinoctial. 

30. The secondaries to the equator, that is, the great circles 
passing through the poles of the equator, are called Meridians, 



DOCTRINE OF THE SPHERE. 13 

because that secondary which passes through the zenith of any 
place is the meridian of that place, and is at right angles both to 
the equator and the horizon, passing as it does through the poles 
of both. (Art. 21.) These secondaries are also called Hour Circles, 
because the arcs of the equator intercepted between them are used 
as measures of time. 

31. The Latitude of a place on the earth, is its distance from 
the equator north or south. The Polar Distance, or angular dis- 
tance from the nearest pole, is the complement of the latitude. 

32. The Longitude of a place is its distance from some stand 
ard meridian, either east or west, measured on the equator. The 
meridian usually taken as the standard, is that of the Observatory 
of Greenwich, near London. If a place is directly on the equator, 
we have only to inquire how many degrees of the equator there 
are between that place and the point where the meridian of Green- 
wich cuts the equator. If the place is north or south of the equa- 
tor, then its longitude is the arc of the equator intercepted between 
the meridian which passes through the place, and the meridian of 
Greenwich. 

33. The Ecliptic is a great circle in which the earth performs 
its annual revolution around the sun. It passes through the center 
of the earth and the center of the sun. It is found by observa- 
tion that the earth does not lie with its axis at right angles to the 
plane of the ecliptic, but that it is turned about 23 degrees out of 
a perpendicular direction, making an angle with the plane itself of 
66^. The equator, therefore, must be turned the same distance 
out of a coincidence with the ecliptic, the two circles making 
an angle with each other of 23, (23 27' 40".) It is particu- 
larly important for the learner to form correct ideas of the eclip- 
tic, and of its relations to the equator, since to these two circles a 
great number of astronomical measurements and phenomena are 
referred. 

34. The Equinoctial Points, or Equinoxes,* are the intersec- 

* The term Equinoxes strictly denotes the times when the sun arrives at the equi- 
noctial points, but it is also frequently used to denote those points themselves. 



14 THE EARTH. 

tions of the ecliptic and equator. The time when the sun crosses 
the equator in returning northward is called the vernal, and in 
going southward, the autumnal equinox. The vernal equinox 
occurs about the 21st of March, and the autumnal the 22d of 
September. 

35. The Solstitial Points are the two points of the ecliptic 
most distant from the equator. The times when the sun comes 
to them are called solstices. The summer solstice occurs about 
the 22d of June, and the winter solstice about the 22d of De- 
cember. 

The ecliptic is divided into twelve equal parts of 30 each, 
called signs, which, beginning at the vernal equinox, succeed each 
other in the following order : 

Northern. Southern. 

1. Aries T 7. Libra ^ 

2. Taurus S 8. Scorpio fll 

3. Gemini EL 9. Sagittarius / 

4. Cancer 2o 10. Capricornus V3 

5. Leo SI 11. Aquarius ~ 

6. Virgo TIE 12. Pisces x 

The mode of reckoning on the ecliptic, is by signs, degrees, 
minutes, and seconds. The sign is denoted either by its name 
or its number. Thus 100 may be expressed either as the 10th 
degree of Cancer, or as 3 s 10 

36. Of the various meridians, two are distinguished by the 
name of Colures. The Equinoctial Colure, is the meridian which 
passes through the equinoctial points. The Solstitial Colure, is 
the meridian which passes through the solstitial points. As the 
solstitial points are 90 from the equinoctial points, so the sol- 
stitial colure is 90 from the equinoctial colure. It is also at right 
angles, or a secondary to both the ecliptic and equator. For like 
every other meridian, it is of course perpendicular to the equator, 
passing through its poles. Moreover, the equinox, being a point 
both in the equator and in the ecliptic, is 90 from the solstice, 
from the pole of the equator, and from the pole of the ecliptic. 



DOCTRINE OF THE SPHERE. 15 

Hence the solstitial colure, which passes through the solstice and 
the pole of the equator, passes also through the pole of the ecHptic, 
being the great circle of which the equinox itself is the pole. 
Consequently, the solstitial colure is a secondary to both the equa- 
tor and the ecliptic. (See Arts. 19, 20, 21.) 

37. The position of a celestial body is referred to the equator 
by its right ascension and declination. (See Art. 27.) Right 
Ascension, is the angular distance from the vernal equinox, meas- 
ured on the equator. If a star is situated on the equator, then its 
right ascension is the number of degrees of the equator between 
the star and the vernal equinox. But if the star is north or south 
of the equator, then its right ascension is the arc of the equator 
intercepted between the vernal equinox and that secondary to the 
equator which passes through the star. Declination is the dis- 
tance of a body from the equator, measured on a secondary to the 
latter. Therefore, right ascension and declination correspond to 
terrestrial longitude and latitude, right ascension being reckoned 
from the equinoctial colure, in the same manner as longitude is 
reckoned from the meridian of Greenwich. On the other hand, 
celestial longitude and latitude are referred, not to the equator, 
but to the ecliptic. Celestial Longitude, is the distance of a body 
from the vernal equinox reckoned on the ecliptic. Celestial Lati- 
tude, is distance from the ecliptic measured on a secondary to the 
latter. Or, more briefly, Longitude is distance on the ecliptic ; 
Latitude, distance from the ecliptic. The North Polar Distance 
of a star, is the complement of its declination. 

38. Parallels of Latitude are small 
circles parallel to the equator. They 
constantly diminish in size as we go 
from the equator to the pole, the ra- 
dius being always equal to the cosine 
of the latitude. In fig. 4, let HO be 
the horizon, EQ the equator, PP the 
axis of the earth, ZN the prime ver- 
tical, and ZL a parallel of latitude of 
any place Z. Then ZE is the lati- 




16 THE EARTH. 

tude, (Art. 31,) and ZP the complement of the latitude ; but Zn 
the radius of the parallel of latitude ZL, is the sine of ZP, and 
therefore the cosine of the latitude. 

39. The' Tropics are the parallels of latitude that pass through 
the solst'ces. The northern tropic is called the tropic of Cancer ; 
the southern, the tropic of Capricorn. 

40. The Polar Circles are the parallels of latitude that pass 
through the poles of the ecliptic, at the distance of 23| degrees 
from the pole of the earth. (Art. 33.) 

41. The earth is divided into five zones. That portion of the 
earth which lies between the tropics, is called the Torrid Zone ; 
that between the tropics and polar circles, the Temperate Zones ; 
and that between the polar circles and the poles, the Frigid 
Zones. 

~* ' * : if : 

42. The Zodiac is the part of the celestial sphere which lies 
about 8 degrees on each side of the ecliptic. This portion of the 
heavens is thus marked off by itself, because the planets are never 
seen further from the ecliptic than this limit. 

43. The elevation of the pole is equal to the latitude of the place. 
The arc PE (Fig. 4.)=ZO.*.PO=ZE which equals the lati- 
tude. 

44. The elevation of the equator is equal to the complement of 
the latitude. 

ZH=90. But ZE=Lat. /. EH=90 Lat.^colatitude. 

45. The distance of any place from the pole (or the polar dis- 
tance) equals the complement of the latitude. 

EP=90. But EZ=Lat..\ZP=90-Lat.=colatitude. 



DIURNAL REVOLUTION. 17 



CHAPTER II. 

DIURNAL REVOLUTION ARTIFICIAL GLOBES ASTRONOMICAL 

PROBLEMS. 

46. THE apparent diurnal revolution of the heavenly bodies 
from east to west, is owing to the actual revolution of the earth 
on its own axis from west to east. If we conceive of a radius of 
the earth's equator extended until it meets the concave sphere of 
the heavens, then as the earth revolves, the extremity of this line 
would trace out a curve on the face of the sky, namely, the celes- 
tial equator. In curves parallel to this, called the circles ofdiurnat 
revolution, the heavenly bodies actually appear to move, every stai 
having its own peculiar circle. After the learner has first rendered 
familiar the real motions of the earth from west to east, he may then, 
without danger of misconception, adopt the common language, 
that all the heavenly bodies revolve around the earth once a day 
from east to west,in circles parallel to the equator and to each other. 

47. The time occupied by a star in passing from any point in 
the meridian until it comes round to the same point again, is called 
a sidereal, day, and measures the period of the earth's revolution 
on its axis. If we watch the returns of the same star from day to 
day, we shall find the intervals exactly equal to one another ; 
that is, the sidereal days are all equal.* Whatever star we select 
for the observation, the same result will be obtained. The stars, 
therefore, always keep the same relative position, and have a 
common movement round the earth, a consequence that natu- 
rally flows from the hypothesis, that their apparent motion is all 
produced by a single real motion, namely, that of the earth. The 
sun, moon, and planets, revolve in like manner, but their returns to 
the meridian are not, like those of the fixed stars, at exactly equal 
intervals. 

48. The appearances of the diurnal motions of the heavenly 

* Allowance is here supposed to be made for the effects of precession, &c. 

3 



THE EARTH. 



bodies are different in different parts of the earth, since every 
place has its own horizon, (Art. 15,) and different horizons are 
variously inclined to each other. Let us suppose the spectator 
viewing the diurnal revolutions, successively, from several different 
positions on the earth. 

49. If he is on the equator, his horizon passes through both poles ; 
for the horizon cuts the celestial vault at 90 degrees in every di- 
rection from the zenith of the spectator ; but the pole is likewise 
90 degrees from his zenith, and consequently, the pole must be 
in his horizon. The celestial equator coincides with his Prime 
Vertical, being a great circle passing through the east and 
west points. Since all the diurnal circles are parallel to the equa- 
tor, they are all, like the equator, perpendicular to his horizon. 
Such a view of the heavenly bodies, is called a right sphere ; or, 

A RIGHT SPHERE is one in which all the daily revolutions of 
the heavenly bodies are in circles perpendicular to the horizon. 

A right sphere is seen only at the equator. Any star situated 
in the celestial equator, would appear to rise directly in the east, at 
noon to pass through the zenith of the spectator, and to set directly in 
the west ; in proportion as stars are at a greater distance from the 
equator towards the pole, they describe smaller and smaller circles, 
until, near the pole, their motion is hardly perceptible. In a right 
sphere every star remains arr equal time above and below the hori- 
zon ; and since the times of their revolutions are equal, the veloci- 
ties are as the lengths of the circles they describe. Consequently, 
as the stars are more remote from the equator towards the pole, 
their motions become slower, until, at the pole, the north star ap- 
pears stationary. 

50. If the spectator advances one degree towards the north 
pole, his horizon reaches one degree beyond the pole of the earth, 
and cuts the starry sphere one degree below the pole of the heav- 
ens, or below the north star, if that be taken as the place of the 
pole. As he moves onward towards the pole, his horizon contin- 
ually reaches further and further beyond it, until when he comes 
to the pole of the earth, and under the pole of the heavens, his 
horizon reaches on all sides to the equator and coincides with it. 



DIURNAL REVOLUTION. 19 

Moreovei, since all the circles of daily motion are parallel to the 
equator, they become, to the spectator at the pole, parallel to the 
horizon. This is what constitutes a parallel sphere. Or, 

A PARALLEL SPHERE is that in which all the circles of daily 
motion are parallel to the horizon. 

51. To render this view of the heavens familiar, the learner 
should follow round in his mind a number of separate stars, one 
near the horizon, one a few degrees above it, and a third near the 
zenith. To one who stood upon the north pole, the stars of the 
northern hemisphere would all be perpetually in view when not 
obscured by clouds or lost in the sun's light, and none of those of 
the southern hemisphere would ever be seen. The sun would 
be constantly above the horizon for six months in the year, and 
the remaining six constantly out of sight. That is, at the pole 
the days and nights are each six months long. The phenomena 
at the south pole are similar to those at the north. 

52. A perfect parallel sphere can never be seen except at one 
of the poles, a point which has never been actually reached by 
man ; yet the British discovery ships penetrated within a few 
degrees of the north pole, and of course enjoyed the view- of a 
sphere nearly parallel. 

53. As the circles of daily motion are parallel to the horizon of 
the pole, and perpendicular to that of the equator, so at all places 
between the two, the diurnal motions are oblique to the horizon. 
This aspect of the heavens constitutes an oblique sphere, which is 
thus defined : 

An OBLIQUE SPHERE is that in which the circles of daily mo- 
tion are oblique to the horizon. 

Suppose for example the spectator is at the latitude of fifty de- 
grees, His horizon reaches 50 beyond the pole of the earth, and 
gives the same apparent elevation to the pole of the heavens. It 
cuts the equator, and all the circles of daily motion, at an angle 
of 40, being always equal to the co-altitude of the pole. Thus, 
let HO (Fig. 5,) represent the horizon, EQ the equator, and 
PP' the axis of the earth. Also, IL mm, &c. parallels of latitude. 



THE EARTH. 




Then the horizon of a spectator Fi g- 5 - 

at Z, in latitude 50 reaches to 
50 beyond the pole (Art. 50) ; 
and the angle ECH, is 40. As 
we advance still further north, 
the elevation of the diurnal cir- 
cles grows less and less, and 
consequently the motions of the 
heavenly bodies more and more 
oblique, until finally, at the pole, 
where the latitude is 90, the 
angle of elevation of the equator 
vanishes, and the horizon and equator coincide with each other 
as before stated. 

54. The CIRCLE OF PERPETUAL APPARITION, is the boundary of 
that space around the elevated pole, where the stars never set 
Its distance from the pole is equal to the latitude of the place. 
For, since the altitude of the pole is equal to the latitude, a star 
whose polar distance is just equal to the latitude, will when at its 
lowest point only just reach the horizon ; and all the stars nearer 
the pole than this will evidently not descend so far as the horizon. 

Thus, mm (Fig. 5,) is the circle of perpetual apparition, be- 
tween which and the north pole, the stars never set, and its dis- 
tance from the pole OP is evidently equal to the elevation of the 
pole, and of course to the latitude. 

55. In the opposite hemisphere, a similar part of the sphere 
adjacent to the depressed pole never rises. Hence, 

The CIRCLE OF PERPETUAL occuLTATioN, is the boundary of that 
space around the depressed pole, within which the stars never rise. 
Thus, m'm' (Fig. 5,) is the circle of perpetual occultation, be- 
tween which and the south pole, the stars never rise. 



56. In an oblique sphere, the horizon cuts the circles of daily 
motion unequally. Towards the elevated pole, more than half 
the circle is above the horizon, and a greater and greater portion 
-is the distance from the equator is increased, until finally, within 



DIURNAL REVOLUTION. 21 

the circle of perpetual apparition, the whole circle is above the 
horizon. Just the opposite takes place in the hemisphere next 
the depressed pole. 40 cordingly, when the sun is in the equator, 
as the equator and horizon, like all other great circles of the 
sphere, bisect each other, the days and nights are equal all over 
the globe. But when the sun is north of the equator, our days 
become longer than our nights, but shorter when the sun is 
south of the equator. Moreover, the higher . the latitude, the 
greater is the inequality in the lengths, of the days and nights. 
All these points will be readily understood by inspecting figure 5. 

57. Most of the phenomena of the diurnal revolution can be 
explained, either on the supposition that the celestial sphere actu- 
ally all turns around the earth once in 24 hours, or that this mo- 
tion of the heavens is merely apparent, arising from the revolu- 
tion of the earth on its axis in the opposite direction, a motion 
of which we are insensible, as we sometimes lose the conscious- 
ness of our own motion in a ship or a steamboat, and observe all 
external objects to be receding from us with a common motion. 
Proofs entirely conclusive and satisfactory, establish the fact, that 
it is the earth and not the celestial sphere that turns ; but these 
proofs are drawn from various sources, and the student is not pre- 
pared to appreciate their value, or even to understand some of 
them, until he has made considerable proficiency in the study of 
astronomy, and become familiar with a great variety of astronom- 
ical phenomena. To such a period of our course of instruction 
we therefore postpone the discussion of the hypothesis of the 
earth's rotation on its axis. 

58. While we retain the same place on the earth, the diurnal 
revolution occasions no change in our horizon, but our horizon 
goes round as well as ourselves. Let us first take our station on 
the equator at sunrise ; our horizon now passes through both the 
poles, and through the sun, which we are to conceive of as at a 
great distance from the earth, and therefore as cut, not by the 
terrestrial but by the celestial horizon. As the earth turns, the 
horizon dips more and more below the sun, at the rate of 15 de- 
grees for every hour, and, as in the case of the polar star, (Art. 50,) 



22 THE ISAltTH. 

the sun appears to rise at the sam'e rate. In six hours, therefore, 
it is depressed 90 degrees below the sun, which brings us directly 
under the sun, which, for our present purjlilse, we may consider as 
having all the while maintained the same fixed position in space. 
The earth continues to turn, and in six hours more, it completely 
reverses the position of our horizon, so that the western part of 
the horizon which at sunrise was diametrically opposite to the 
sun now cuts the sun, and soon afterwards it rises above the level 
of the sun, and the sun sets. During the next twelve hours, the 
sun continues on the invisible side of the sphere, until the hori- 
zon returns to the position from which it started, and a new day 
begins. 

59. Let us next contemplate the similar phenomena at the poles. 
Here the horizon, coinciding as it does with the equator, would 
cut the sun through its center, and the sun would appear to re- 
volve along the surface of the sea, one half above and the other 
half below the horizon. This supposes the sun in its annual 
revolution to be at one of the equinoxes. When the sun is north 
of the equator, it revolves continually round in a path which, 
during a single revolution, appears parallel to the equator, and it 
is constantly day ; and when the sun is south of the equator, it is, 
for the same reason, continual night. 

60. We have endeavored to conceive of the manner in which 
the apparent diurnal movements of the sun are really produced at 
two stations, namely, in the right sphere, and in the parallel sphere. 
These two cases being clearly understood, there will be little dif- 
ficulty in applying a similar explanation to an oblique sphere 

ARTIFICIAL GLOBES. 

61. Artificial globes are of two kinds, terrestrial and celestial. 
The first exhibits a miniature representation of the earth ; the 
second, of the visible heavens ; and both show the various circles 
by which the two spheres are respectively traversed. Since all 
globes are similar solid figures, a small globe, imagined to be sit- 
uated at the center of the earth or of the celestial vault, may rep- 



ARTIFICIAL GLOBES. 23 

resent all the visible objects and artificial divisions of either sphere, 
and with great accuracy and just proportions, though on a scale 
greatly reduced. The study of artificial globes, therefore, cannot 
be too strongly recommended to the student of astronomy.* 

62. An artificial globe is encompassed from north to south by 
a strong brass ring to represent the meridian of the place. This 
ring is made fast to the two poles and thus supports the globe, 
while it is itself supported in a vertical position by means of a 
frame, the ring being usually let into a socket in which it may be 
easily slid, so as to give any required elevation to the pole. The 
brass meridian is graduated each way from the equator 'to the 
pole 90, to measure degrees of latitude or declination, according 
as the distance from the equator refers to a point on the earth or 
in the heavens. The horizon is represented by a broad zone, made 
broad for the convenience of carrying on it a circle of azimuth, an- 
other of amplitude, and a wide space on which are delineated the 
signs of the ecliptic, and the sun's place for every day in the year ; 
not because these points have any special connexion with the hori- 
zon, but because this broad surface furnishes a convenient place 
for recording them. 

63. Hour Circles are represented on the terrestrial globe by 
great circles drawn through the pole of the equator ; but, on the 
celestial globe, corresponding circles pass through the poles of the 
ecliptic, constituting circles of celestial latitude, (Art. 37,) while the 
brass meridian, being a secondary to the equinoctial, becomes an 
hour circle of any star which, by turning the globe, is brought un- 
der it. 

64. The Hour Index is a small circle described around the pole 
of the equator, on which are marked the hours of the day. As 
this circle turns along with the globe, it makes a complete revo- 
lution in the same time with the equator ; or, for any less period. 

* It were desirable, indeed, that every student of the science should have the celes- 
tial globe at least, constantly before him. One of a small size, as eight or nine inches, 
will answer the purpose, although globes of these dimensions cannot usually be :elied 
on for nice measurements. 



24 THE EARTH. 

the same number of degrees of this circle and of the equator pass 
under the meridian. Hence the hour index measures arcs of 
right ascension. (Art. 37.) 

65. The Quadrant of Altitude is a flexible strip of brass, gradu- 
ated into ninety equal parts, corresponding in length to degrees 
on the globe, so that when applied to the globe and bent so as 
closely to fit its surface, it measures the angular distance between 
any two points. When the zero, or the point where the gradua- 
tion begins, is laid on the pole of any great circle, the 90th degree 
will reach to the circumference of that circle, and being therefore 
a great circle passing through the pole of another great circle, it 
becomes a secondary to the latter. (Art. 21.) Thus the quadrant 
of altitude may be used as a secondary to any great circle on the 
sphere ; but it is used chiefly as a secondary to the horizon, the 
point marked 90 being screwed fast to the pole of the horizon, 
that is, the zenith, and the other end, marked 0, being slid along 
between the surface^of the sphere and the wooden horizon. It 
thus becomes a vertical circle, on which to measure the altitude 
of any star through which it passes, or from which to measure 
the azimuth of the star, which is the arc of the horizon intercept- 
ed between the meridian and the quadrant of altitude passing 
through the star, (Art. 27.) 

66. To rectify the globe for any place, the north pole must be 
elevated to the latitude of the place (Art. 43) ; then the equator 
and all the diurnal circles will have their due inclination in respect 
to the horizon ; and, on turning the globe, (the celestial globe west, 
and the terrestrial east,) every point on either globe will revolve as 
the same point does in nature ; and the relative situations of all 
places will be the same as on the respective native spheres. 

PROBLEMS ON THE TERRESTRIAL GLOBE. 

67. To find the Latitude and Longitude of a place : Turn the 
globe so as to bring the place to the brass meridian ; then the de- 
gree and minute on the meridian directly over the place will indi- 
cate its latitude, and the point of the equator under the meridian, 
will show its longitude. 



PROBLEMS ON THE TERRESTRIAL GLOBE. 25 

Ex. What are the Latitude and Longitude of the city of New 
York 7 ? 

68. To find a place having its latitude and longitude given : Bring 
to the brass meridian the point of the equator corresponding to 
the longitude, and then at the degree of the meridian denoting the 
latitude, the place will be found. 

Ex. What place on the globe is in Latitude 39 N. and Longi- 
tude 77 W. ? 

69. To find the bearing and distance of two places: Rectify the 
globe for one of the places (Art. 66) ; screwrfhe quadrant of alti- 
tude to the zenith,* and let it pass through the other place. Then 
the azimuth will give the bearing of the second place from the 
first, and the number of degrees on the quadrant of altitude, mul- 
tiplied by 69|, (the number of miles in a degree,) will give the 
distance between the two places. 

Ex. What is the bearing of New Orleans from New York, and 
what is the distance between these places ? 

70. To determine the difference of time in different places : 
Bring the place that lies eastward of the other to the meridian, 
and set the hour index at XII. Turn the globe eastward until 
the other place comes to the meridian, then the index will point 
to the hour required. 

Ex. When it is noon at New York, what time is it at London ? 

71. The hour being given at any place, to tell what hour it is in 
any other part of the world : Find the difference of time between 
the two places, (Art. 70,) and, if the place whose time is required 
is eastward of the other, add this difference to the given time, but 
if westward, subtract it. 

Ex. What time is it at Canton, in China, when it is 9 o'clock 
A. M. at New York? 

72. To find the antceci,^ the periceci,^. and the antipodes^ of any 

* The zenith will of course be the point of the meridian over the place, 
t avn OIKOS. J irspi oi/coj. $ aim TKf. 

4 



26 THE EARTH. 

place : Bring the given place to the meridian ; then, under the 
meridian, in the opposite hemisphere, in the same degree of lati- 
tude, will be found thi antoeci. The same place remaining under 
the meridian, set the index to XII, and turn the globe until the 
other XII is under the index ; then the perio3ci will be on the me- 
ridian, under the same degree of latitude with the given place, 
and the antipodes will be under the meridian, in the same latitude, 
in the opposite hemisphere. 

Ex. Find the antoeci, the perioeci, and the antipodes of the citi- 
zens of New York. 

The antoeci have the same hour of the day, but different seasons 
of the year ; the perieeci have the same seasons, but opposite hours ; 
and the antipodes have both opposite hours and opposite seasons. 

73. To rectify the globe for the sun's place : On the wooden 
horizon, find the day of the month, and against it is given the sun's 
place in the ecliptic, expressed by signs and degrees.* Look for 
the same sign and degree on the ecliptic, bring that point to the 
meridian and set the hour index to XII. To all places under the 
meridian it will then be noon. 

Ex. Rectify the globe for the sun's place on the 1st of September. 

74. The latitude of the place being given, to find the time of the 
sun's rising and setting on any given day at that plaice : Having 
rectified the globe for the latitude, (Art. 66,) bring the sun's place 
in the ecliptic to the graduated edge of the meridian, and set the 
hour index to XII ; then turn the globe so as to bring the sun to 
the eastern and then to the western horizon, and the hour index 
will show the times of rising and setting respectively. 

Ex. At what time will the sun rise and set at New Haven, 
Lat. 41 18', on the 10th of July? 

PROBLEMS ON THE CELESTIAL GLOBE. 

75. To find the Declination and Right Ascension of a heavenly 
body : Bring the place of the body (whether the sun or a star) to 
the meridian. Then the degree and minute standing over it will 

* The larger globes have the dqy of the month marked against the corresponding 
sign on the ecliptic itself. 



PROBLEMS O? THE CELESTIAL GLOBE. 27 

show its declination, and the point of the equinoctial under the 
meridian will give its right ascension. It will be remarked, that 
the decimation and right ascension are found in the same manner 
as latitude and longitude on the terrestrial globe. Right ascen- 
sion is expressed either in degrees or in hours ; both being reck- 
oned from the vernal equinox, (Art. 37.) 

Ex. What is the declination and right ascension of the bright 
star Lyra ? also of the sun on the 5th of June 1 

76. To represent the appearance of the heavens at any time: 
Rectify the globe for the latitude, bring the sun's place in the 
ecliptic to the meridian, and set the hour index to XII ; then turn 
the globe westward until the index points to the given hour, and 
the constellations would then have the same appearance to an eye 
situated at the center of the globe, as they have at that moment 
in the sky. 

Ex. Required the aspect of the stars at New Haven, Lat. 41 
18', at 10 o'clock, on the evening of December 5th. 

77. To find the altitude and azimuth of any star : Rectify the 
globe for the latitude and the sun's place, and let the quadrant 
of altitude be screwed to the zenith, and be made to pass through 
the star. The arc on the quadrant, from the horizon to the star, 
will denote its altitude, and the arc of the horizon from the 
meridian to the quadrant, will be its azimuth. 

Ex. What are the altitude and azimuth of Sirius (the brightest 
of the fixed stars) on the 25th of December at 10 o'clock in the 
evening, in Lat. 41? 

78. To find the angular distance of two stars from each other : 
Apply the zero mark of the quadrant of altitude to one of the 
stars, and the point of the quadrant which falls on the other star, 
will show the angular distance between the two. 

Ex. What is the distance between the two largest stars of the 
Great Bear?* 

* These two stars are sometimes called " the Pointers," from the line which passei 
through them being always nearly in the direction of the north star. The angular 
distance between them is about 5, and may be learned as a standard for reference in 
estimating, by the eye, the distance between any two points on the celestial vault. 



28 PARALLAX. 

79. To find the sun's meridian altitude, the latitude and day 
of the month being given: Having rectified the globe for the 
latitude, (Art. 66,) bring the sun's place in the ecliptic to the me- 
ridian, and count the number of degrees and minutes between 
that point of the meridian and the zenith. The complement of 
this arc will be the sun's meridian altitude. 

Ex. What is the sun's meridian altitude at noon on the 1st of 
August, in Lat. 41 18'? 



CHAPTER III. 

OF PARALLAX, REFRACTION, AND TWILIGHT. 

80. PARALLAX is the apparent change of place which bodies 
undergo by being viewed from different points\ Thus in figure 
6, let A represent the earth, CH' the horizon, H'Z a quadrant of 



Fig. 6 




a great circle of the heavens, extending from the horizon to the 
zenith ; and let E, F, G, H, be successive positions of the moon 
at different elevations, from the horizon to the meridian. Now a 
spectator on the surface of the earth at A, would refer the place 
of E to h, whereas, if seen from the center of the earth, it would 



ARALLAX. 29 

appear at H'. The arc H'h is called the parallactic arc, and the 
angle H'EA, or its equal AEC, is the angle of parallax. The 
same is true of the angles at F, G, and H, respectively. 

81. Since then a heavenly body is liable to be referred to dif- 
ferent points on the celestial vault, when seen from different parts 
of the earth, and thus some confusion occasioned in the deter- 
mination of points on the celestial sphere, astronomers have agreed 
to consider the true place of a celestial object to be that where it 
would appear if seen from the center of the earth. The doctrine 
of parallax teaches how to reduce observations made at any place 
on the surface of the earth, to such as they would be if made 
from the center. 

82. The angle AEC is called the horizontal parallax, which 
may be thus defined. Horizontal Parallax, is the change of po- 
sition which a celestial body, appearing in the horizon as seen 
from the surface of the earth, would assume if viewed from the 
earth's center. It is the angle subtended by the semi-diameter 
of the earth, as viewed from the body itself. If we consider any 
one of the triangles represented in the figure, ACG for example, 

Sin. AGO : Sin. GAZ : : AC : CG 
,. Sin. Parallax^ S 



CG 

Hence the sine of the angle of parallax, or (since the angle of 
parallax is always very small*) the parallax itself varies as, the 
sine of the zenith distance of the body directly, and the distance 
of the body from the center of the earth inversely. Parallax, there- 
fore, increases as a body approaches the horizon, (but increasing 
with the sines, it increases much slower than in the simple ratio 
of the distance from the zenith,) and diminishes, JLS the distance 
from the spectator increases. Again, since the parallax AGC is as 
the sine of the zenith distance, let P represent the horizontal par- 
allax, and P 7 the parallax at any altitude ; then, 

* The moon, on account of its nearness to the earth, has the greatest horizontal 
parallax of any of the heavenly bodies ; yet this is less than 1' (being 57*) while the 
greatest parallax of any of the planets does not exceed 30". The difference in an 
arc of 1, between the length of the arc and the sine, is only O."18. 



30 THE EARTH. 

P' 



P' : P::sin. zenith dist.: sin. 90/.P=- 



sin. zen. dist. 

Hence, the horizontal parallax of a body equals its parallax at 
any altitude, divided by the sine of its distance from the zenith. 

83. From observations, therefore, on the parallax of a body at 
any elevation, we are enabled to find the angle subtended by the 
semi-diameter of the earth as seen from the body. Or if the 
norizontal parallax is given, the parallax at any altitude may be 
found, for 

P/=Pxsin. zenith distance. 

Hence, in the zenith the parallax is nothing, and is at its max- 
imum in the horizon. 

84. It is evident from the figure, that the effect of parallax 
upon the place of a celestial body is to depress it. Thus, in con- 
sequence of parallax, E is depressed by the arc H'h ; F by the 
arc Pp ; G by the arc Rr ; while H sustains no change. Hence, 
in all calculations respecting the altitude of the sun, moon, or plan- 
ets, the amount of parallax is to be added ; the stars, as we shall 
see hereafter, have no sensible parallax. As the depression which 
arises from parallax is in the direction of a vertical circle, a body, 
when on the meridian, has only a parallax in declination ; but 
in other situations, there is at the same time a parallax in 
declination and right ascension ; for its direction being oblique 
to the equinoctial, it can be resolved into two parts, one of which 
(the declination) is perpendicular, and the other (the right ascen- 
sion) is parallel to the equinoctial. 

85. The mode of determining the horizontal parallax, is as 
follows : 

Let O, O', (Fig. 7,) be two places on the earth, situated under 
the same* meridian, at a great distance from each other ; one place, 
for example, at the Cape of Good Hope, and the other in the north 
of Europe. The latitude of each place being known, the arc of 
the meridian OO' is known, and the angle OCO' also is known. 
Let the celestial body M, (the moon for example,) he observed 
simultaneously at O and O', and its zenith distance at each place 



PARALLAX. 



31 



accurately taken, namely, ZY and 

Z'Y' ; then the angles ZOM and 

Z'O'M, and of course their sup- 

plements COM,CO'M are found. 

Then in the quadrilateral figure 

COMO', we have all the angles 

and the two radii, CO, CO', 

whence by joining OO', the side 

OM may be easily found. Hav- 

ing CO and OM, we may find 

CMO=sine of the angle of par- 

allax ; or (since the arc is very 

small) equals the parallax P'. 

But when M as seen from O is in the horizon, ZOM becomes a 

right angle, and its sine equal to radius. Then, CM being found, 




CM : CO : : 1 : P=horizontal parallax=^. 

On this principle, the horizontal parallax of the moon was de- 
termined by La Caille and La Lande, two French astronomers, 
one stationed at the Cape of Good Hope, the other at Berlin ; and 
in a similar way the parallax of Mars was ascertained, by ob- 
servations made simultaneously at the Cape of Good Hope and 
at Stockholm. 

86. On account of the great distance of the sun, his horizontal 
parallax, which is only 8".6, cannot be accurately ascertained by 
this method. It can, however, be determined by means of the 
transits of Venus, a process to be described hereafter. 

87. The determination of the horizontal parallax of a celestial 
body is an element of great importance, since it furnishes the 
means of estimating the distance of the body from the center of 
the earth. Thus, if the angle AEC (Fig. 6,) be found, the radius 
of the earth AC being known, we have in the triangle AEC, 
right angled at A, the side AC and all the angles, to find the hypo- 
thenuse CE, which is the distance of the moon from the center 
of the earth. 



32 THE EARTH. 

REFRACTION. 

88. While parallax depresses the celestial bodies subject to it, 
refraction elevates them ; and it affects alike the most distant 
as well as nearer bodies, being occasioned by the change of di- 
rection which light undergoes in passing through the atmos- 
phere. Let us conceive of the atmosphere as made up of a great 
number of concentric strata, as A A, BB, CC, and DD, (Fig. 8,) 

Fig. 8. 




increasing rapidly in density (as is known to be the fact) in ap- 
proaching near to the surface of the earth. Let S be a star, from 
which a ray of Ifght S enters the atmosphere at , where, being 
turned towards the radius of the convex surface, it would change 
its direction into the line ab, and again into be, and cO, reach- 
ing the eye at O. Now, since an object always appears in the 
direction in which the light finally strikes the eye, the star would 
be seen in the direction of the last ray cO, and the star would 
apparently change its place, in consequence of refraction, from 
S to S', being elevated out of its true position. Moreover, 
since on account of the constant increase of density in descend- 
ing through the atmosphere, the light would be continually turned 
out of its course more and more, it would therefore move, not 
in the polygon represented in the figure, but in a corresponding 
curve, whose curvature is rapidly increased near the surface of 
the earth. 

89. When a body is in the zenith, since a ray of light from it 
enters the atmosphere at right angles to the refracting medium, it 
suffers no refraction. Consequently, the position of the heavenly 



REFRACTION. 



33 



bodies, when in the zenith, is not changed by refraction, while, 
near the horizon, where a ray of light strikes the medium very 
obliquely, and traverses the atmosphere through its densest part, 
the refraction is greatest. The following numbers, taken at dif- 
ferent altitudes, will show how rapidly refraction diminishes from 
the horizon upwards. The amount of refraction at the horizon 
is 3 00". At different elevations it is as follows. 



Elevation. 


Refraction. 


Elevation. 


Refraction. 


10' 


32' 00" 


30 


1' 40" 


20 


30 00 


40 


1 09 


1 00 


24 25 


45 


58 


5 00 


10 00 


60 


33 


10 00 


5 20 


80 


10 


20 00 


2 39 


90 


00 



F/om this table it appears, that while refraction at the horizon 
is 34 minutes, at so small an elevation as only 10 minutes above 
the horizon it loses 2 minutes, more than the entire change from 
the elevation of 30 to the zenith. From the horizon to 1 above, 
the refraction is diminished nearly 10 minutes. The amount at 
the horizon, at 45, and at 90, respectively, is 34', 58", and 0. In 
finding the altitude of a heavenly body, the effect of parallax must 
be added, but that of refraction subtracted. 

90. Let us now learn the method, by which the amount of re- 
fraction at different elevations is ascertained. To take the sim- 
plest case, we will suppose ourselves in a high latitude, where 
some of the stars within the circle of perpetual apparition pass 
through the zenith of the place. We measure the distance of 
such a star from the pole when on the meridian above the pole, 
that is, in the zenith, where it is not at all affected by refraction, 
and again its distance from the pole in its lower culmination. 
Were it not for refraction, these two polar distances would be 
equal, since, in the diurnal revolution of a star, it is in fact always 
at the same distance from the pole ; but, on account of refraction, 
the lower distance will be less than the upper, and the difference 
between the two will show the amount of refraction at the lower 
culmination, the latitude of the place being known. 

Example. At Paris, latitude 48 50', a star was observed to 
5 



34 



THE EARTH. 



pass the meridian 6' north of the zenith, and consequently, 41 4', 
from the pole.* It should have passed the meridian at the same 
distance below the pole, but the distance was found to be only 
40 57' 35". Hence, 41 4'-40 57' 35"=6' 25" is the refraction 
due to that altitude, that is, at the altitude of 7 46'=(48 50'-- 
41 4'). By taking similar observations in various places situated 
in high latitudes, the amount of refraction might be ascertained 
for a number of different altitudes, and thus the law of increase 
in refraction as we proceed from the zenith towards the horizon, 
might be ascertained. - . ^ 

91. Another method of finding the refraction at different alti- 
tudes, is as follows. Take the altitude of the sun or a star, whose 
right ascension and declination are known, and note the time by 
the clock. Observe also when it crosses the meridian, and the 
difference of time between the two observations will give the hour 
angle ZP#, (Fig. 9.) In this triangle ZPx we also know PZ the 

Fig. 9. 




co-latitude and Pa? the co-declination. Hence we can find the co- 
altitude Z#, and of course the true altitude. Compare the alti- 
tude thus found with that before determined by observation, and 
the difference will be the refraction due to the apparent altitude. 

*For the polar distance of the place=90-48 50'=41 10'; and 41 10'-6'= 
41 4. 



REFRACTION. 35 

Ex. On May 1, 1738, at 5h. 20m. in the morning, Cassini ob- 
served the altitude of the sun's center at Paris to be 5 0' 14". The 
latitude of Paris being 48 50' 10", and the sun's declination at 
that time being 15 0' 25" : Required the refraction. 

By spherical trigonometry, Zx will be found equal to 85 10' 
8"; consequently, the true altitude was 4 49' 52". Now to 5 
0' 14", the apparent altitude, 9" must be added for parallax, 
and we have 5 0' 23" the apparent altitude corrected for parallax 
Hence, 5 0' 23"-4 49' 52"=10' 31", the refraction at the ap- 
parent altitude 5 0' 14".* 

92. By these and similar methods, we could easily determine 
the refraction due to any elevation above the horizon, provided 
the refracting medium (the atmosphere) were always uniform. 
But this is not the fact : the refracting power of the atmosphere 
is altered by changes in density and temperature.f Hence in 
delicate observations, it is necessary to take into the account the 
state of the barometer and of the thermometer, the influence of 
the variations of each having been very carefully investigated, 
and rules having been given accordingly. With every precaution 
to insure accuracy, on account of the variable character of the 
refracting medium, the tables are not considered as entirely accu- 
rate to a greater distance from the zenith than 74 ; but almost all 
astronomical observations are made at a greater altitude than this. 

93. Since the whole amount of refraction near the horizon ex- 
ceeds 33', and the diameters of tjie sun and moon are severally 
less than this, these luminaries are in view both before they have 
actually risen and after they have set. 

94. The rapid increase of refraction near the horizon, is strik- 
ingly evinced by th'e oval figure which the sun assumes when 
near the horizon, and which is seen to the greatest advantage 
when light clouds enable us to view the solar disk. Were all 

* Gregory's Ast. p. 65. 

t It is said that the effects of humidity are insensible ; for the most accurate 
experiments seem to prove that watery vapor diminishes the density of air in the 
same ratio as its own refractive power is greater than that of air. (New Encyc. 
Brit. Ill, 762.) 



36 THE EARTH. 

parts of the sun equally raised by refraction, there would be no 
change of figure ; but since the lower side is more refracted than 
the upper, the effect is to shorten the vertical diameter and thus 
to give the disk an oval form. This effect is particularly remark- 
able when the sun, at his rising or setting, is observed from the 
top of a mountain, or at an elevation near the sea shore ; for in 
such situations the rays of light make a greater angle than or- 
dinary with a perpendicular to the refracting medium, and the 
amount of refraction is proportionally greater. In some cases of 
this kind, the shortening of the vertical diameter of the sun has 
been observed to amount to 6', or about one fifth of the whole.* 

95. The apparent enlargement of the sun and moon in the hori- 
zon, arises from an optical illusion. These bodies in fact are 
not seen under so great an angle when in the horizon, as when on 
the meridian, for they are nearer to us in the latter case than in 
the former. The distance of the sun is indeed so great that it 
makes very "little difference in his apparent diameter, whether he 
is viewed in the horizon or on the meridian ; but with the moon 
the case is otherwise ; its angular diameter, when measured with 
instruments, is perceptibly larger at the time of its culmination. 
Wlfy then do the sun and moon appear so much larger when near 
the horizon ? It is owing to that general law, explained in optics, 
by which we judge of the magnitudes of distant objects, not 
merely by the angle they subtend at the eye, but also by our im- 
pressions respecting their distance, allowing, under a given angle, 
* a greater magnitude as we imagine the distance of a body to be 
greater. Now, on account of the numerous objects usually in 
sight between us and the sun, when on the horizon, he appears 
much further removed from us than when on the meridian, and 
we assign to him a proportionally greater magnitude. If we view 
the sun, in the two positions, through smoked glass, no such dif- 
ference of size is observed, for here no objects are seen but the 
sun himself. 



* In extreme cold weather, this shortening of the sun's vertical diameter sometimes 
exceeds this amount. 



TWILIGHT. 



37 



TWILIGHT. 

96. Twilight also is another phenomenon depending upon the 
agency of the earth's atmosphere. It is due partly to refraction 
and partly to reflexion, but mostly the latter. While the sun 
is within 18 of the horizon, before it rises or after it sets, some 
portion of its light is conveyed to us by means of numerous re- 
flections from the atmosphere. Let AB (Fig. 10,) be the horizon 

Fig. 10. 




of the spectator at A, and let SS be a ray of light from the sun 
when it is two or three degrees below the horizon. Then to 
the observer at A, the segment of the atmosphere ABS would be 
illuminated. To a spectator at C, whose horizon was CD, the 
small segment SD# would be the twilight ; while, at E, the twi- 
light would disappear altogether. 

97. At the equator, where the circles of daily motion are per- 
pendicular to the horizon, the sun descends through 18 in an 
hour and twelve minutes (}= Ijh.), and the light of day there- 
fore declines rapidly, and as rapidly advances after daybreak in the 
morning. At the pole, a constant twilight is enjoyed while the sun 
is within 18 of the horizon, occupying nearly two thirds of the 
half year when the direct light of the sun is withdrawn, so that 
the progress from continual day to constant night is exceedingly 
gradual. To the inhabitants of an oblique sphere, the twilight 
is longer in proportion as the place is nearer the elevated pole. 

98. Were it not for the power the atmosphere has of dispersing 



30 THE EARTH. 

the solar light, and scattering it in various directions, no objects 
would be visible to us out of direct sunshine ; every shadow of a 
passing cloud would be pitchy darkness ; the stars would be visi- 
ble all day, and every apartment into which the sun had not di- 
rect admission, would be involved in the obscurity of night. This 
scattering action of the atmosphere on the solar light, is greatly 
increased by the irregularity of temperature caused by the sun, 
which throws the atmosphere into a constant state of undulation, 
and by thus bringing together masses of air of different tempera- 
tures, produces partial reflections and refractions at their common 
boundaries, by which means much light is turned aside from the 
direct course, and diverted to the purposes of general illumination.* 
In the upper regions of the atmosphere, as on the tops of very 
high mountains, where the air is too much rarefied to reflect much 
light, the sky assumes a black appearance, and stars become visi- 
ble in the day time. 



CHAPTER IV. 

OF TIME. 

99. TIME is a measured portion of indefinite duration. 

Any event may be taken as a measure of time, which divides 
a portion of duration into equal parts ; as the pulsations of the 
wrist, the vibrations of a pendulum, or the passage of sand from 
one vessel into another, as in the hour-glass. 

100. The great standard of time is the period of the revolution 
of the earth on its axis, which, by the most exact observations, is 
found to be always the same. The time of the earth's revolution 
on its axis is called a sidereal day, and is determined by the revo- 
lution of a star from the instant it crosses the meridian, until it 
comes round to the meridian again. This interval being called a 

'> Herschel. 



TIME. 39 

sidereal day, it is divided into 24 sidereal hours. Observations 
taken upon numerous stars, in different ages of the world, show 
that they all perform their diurnal 'revolutions in the same time, 
and that their motion during any part of the revolution is per- 
fectly uniform. 

101. Solar time is reckoned by the apparent revolution of the 
sun, from the meridian round to the same meridian again. Were 
the sun stationary in the heavens, like a fixed star, the time of its 
apparent revolution would be equal to the revolution of the earth 
on its axis, and the solar and the sidereal days would be equal. 
But since the sun passes from west to east, through 360 in 365 
days, it moves eastward nearly 1 a day, (59' 8". 3). While, 
therefore, the earth is turning round on its axis, the sun is moving 
in the same direction, so that when we have come round under 
the same celestial meridian from which we started, we do not 
find the sun there, but he has moved eastward nearly a degree, 
and the earth must perform so much more than one complete 
revolution, in order to come under the sun again. Now since a 
place on the earth gains 359 in 24 hours, how long will it take 
to gain 1 ? 

94. 

359 : 24 : : 1 : _=r4 m nearly. 
359 

Hence the solar day is about 4 minutes longer than the sidereal ; 
and if we were to reckon the sidereal day 24 hours, we should 
reckon the solar day 24h. 4m. To suit the purposes of society at 
large, however, it is found most convenient to reckon the solar day 
24 hours, and to throw the fraction into the sidereal day. Then, 

24h. 4m. : 24 : : 24 : 23h. 56m. (23h. 56 m 4 8 .09) = the length 
of a sidereal day. 

102. The solar days, however, do not always differ from the 
sidereal by precisely the same fraction, since the increments of 
right ascension, (Art. 37,) which measure the difference between 
a sidereal and a solar day, are not equal to each other. Apparent 
time, is time reckoned by the revolutions of the sun from the 
meridian to the meridian again. These intervals being unequal, 
of course the apparent solar days are unequal to each other. 



40 THE EARTH. 

103. Mean time, is time reckoned by the average length of all 
the solar days throughout the year. This is the period which con- 
stitutes the civil day of 24 hours, beginning when the sun is on 
the lower meridian, namely, at 12 o'clock at night, and counted 
by 12 hours from the lower to the upper culmination, and from 
the upper to the lower. The astronomical day is the apparent 
solar day counted through the whole 24 hours, instead of by pe- 
riods of 12 hours each, and begins at noon. Thus 10 days and 
14 hours of astronomical time, would be 11 days and 2 hours of 
civil time. 

104. Clocks are usually regulated so as to indicate mean solar 
time ; yet as this is an artificial period, not marked off, like the 
sidereal day, by any natural event, it is necessary to know how 
much is to be added to or subtracted from the apparent solar 
time, in order to give the corresponding mean time. The inter- 
val by which apparent time differs from mean time, is called the 
equation of time. If a clock were constructed (as it may be) so 
as to keep exactly with the sun, going faster or slower according 
as the increments of right ascension were greater or smaller, and 
another clock were regulated to mean time, then the difference 
of the two clocks, at any period, would be the equationof time 
for that moment. If the apparent clock were faster than the 
mean, then the equation of time must be subtracted ; but if the 
apparent clock were slower than the mean, then the equation of 
time must be added, to give the mean time. The two clocks 
would differ most about the 3d of November, when the apparent 
time is 16 m greater than the mean (16 m 17 s ). But, since appa- 
rent time is sometimes greater and sometimes less than mean 
time, the two must obviously be sometimes equal to each other. 
This is in fact the case four times a year, namely, April 15th, 
June 15th, September 1st, and December 22d. These epochs, 
however, do not remain constant ; for, on account of the change 
in the position of the perihelion, or the point where the earth is 
nearest the sun, (which shifts its place from west to east about 
12" a year,) the period when the sun's motions are most rapid, as 
well as that when they are slowest, will occur at different parts of 
the year. The change is indeed exceedingly small in a single 



TIME. 



41 



year ; but in the progress of ages, the time of year when the sun's 
motion in its orbit is most accelerated, will not be, as at present, on 
the first of January, but may fall on the first of March, June, or 
any other day of the year, and the amount of the equation of 
time is obviously affected by the sun's distance from its perihelion, 
since the sun moves most rapidly when nearest the perihelion, and 
slowest when furthest from that point. 

105. The inequality of the solar days depends on two causes, the 
unequal motion of the earth in its orbit, and the inclination of the 
equator to the ecliptic. 

First, on account of the eccentricity* of the earth's orbit, the 
earth actually moves faster from the autumnal to the vernal equi- 
nox, than from the vernal to the autumnal, the difference of the 
two periods being about eight days (7d. 17h. 17m.) Thus, let 

Kg. 11. 




AEB (Fig. 11,) represent the earth's orbit, S being the place of 



* The exact figure of the earth's orbit will be more particularly shown hereafter. 
All that the student requires to know, in order to understand the present subject, 



42 THE EARTH. 



the sun, A the perihelion, or nearest distance of the earth from 
the sun, B the aphelion, or greatest distance, and E, E', E f/ , posi- 
tions of the earth in different points of its orbit. The place of 
the earth among the signs is the part of the heavens to which it 
would be referred if seen from the sun ; and the place of the sun 
is the part of the heavens to which it is referred as seen from the 
earth. Thus, when the earth is at E, it is said to be in Aries ; 
and as it moves from E through E' to A, its path in the heavens 
is through Aries, Taurus, Gemini, &c. Meanwhile the sun takes 
its place successively in Libra, Scorpio, Sagittarius, &'c. Now, 
as will be shown more fully hereafter, the earth moves faster 
when proceeding from Aries through its perihelion to Libra, than 
from Libra through its aphelion to Aries, and, consequently, de- 
scribes the half of its apparent orbit in the heavens, T, S>, ^, 
sooner than the half =*=, V3, T. The line of the apsides, that is, 
the major axis of the ellipse, is so situated at present, that the 
perihelion is in the sign Cancer, nearly 100 (99 30' 5") from the 
vernal equinox. The earth passes through it about the first of 
January, and then its velocity is the greatest in the whole year, 
being always greater as the distance is less, the angular velocity 
being inversely as the square of the distance, as will be shown by 
and by. 

106. But differences of time are not reckoned on the eclip- 
tic, but on the equinoctial ; for the ecliptic being oblique to the 
meridian in the diurnal motion, and cutting it at different angles at 
different times, equal portions will not pass under the meridian in 
equal times, and therefore such portions could not be employed, as 
they are in the equinoctial, as measures of time. If therefore the 
sun moved uniformly in his orbit, so as to make the daily incre- 
ments of longitude equal, still the corresponding arcs of right as- 
cension, which determine the lengths of the solar day, would be 
unequal. Let us start from the equinox, from which both longi- 
tude and right ascension are reckoned, the former on the ecliptic, 



is that the earth's orbit is an ellipse, and that the earth's real motion, and conse- 
quently the sun's apparent motion, is greater in proportion as the earth is nearer 
the sun. 



TIME. 



43 



the latter on the equinoctial. Suppose the sun has described 70 
of longitude ; then to ascertain the corresponding arc of right as- 
cension, we let a meridian pass through the sun : the point where 
it cuts the equator gives the sun's right ascension. Now since the 
ecliptic makes an acute angle with the meridian, while the equi- 
noctial makes a right angle with it, consequently the arc of longi- 
tude is greater than the arc of right ascension. The difference, 
however, grows constantly less and less as we approach the tropic, 
as the angle made between the ecliptic and the meridian constantly 
incre'ases, until, when we reach the tropic, the meridian is at right 
angles to both circles, and the longitude and right ascension each 
equals 90, and they are of course equal to each other. ' Beyond 
this, from the tropic to the other equinox, the arc of the ecliptic 
intercepted between the meridian and the autumnal equinox being 
greater than the corresponding arc of the equinoctial, of course 
its supplement, which measures the longitude, is less than the sup- 
plement of the corresponding arc of the equator which measures 
the right ascension. At the autumnal equinox again, the right 
ascension and longitude become equal. In a similar manner we 
might show that the daily increments of longitude and right as- 
cension are unequal. 

In order to illustrate the foregoing points, let T & (Fig. 12,) 

Fig. 12. 




represent the equator, T T =^= the ecliptic, and PSE, PS'E', two 
meridians meeting the sun in S and S'. Then in the triangle YES, 



44 THE EARTH. 

the arc of longitude TS, is greater than TE, the corresponding 
arc of right ascension; but towards the tropic the difference 
between the two arcs evidently grows less and less, until at T 
the arcs become equal, being each 90. But, beyond the tropic, 
since TE'=a=, TS'=*=, are equal to each other, each being equal 
to 180, and since S'=*= is greater than E'^, therefore TS' must 
be less than TE'. 

1 07. As the whole arc of right ascension reckoned from the 
first of Aries, does not keep uniform pace with the corresponding 
arc of longitude, so the daily increments of right ascension differ 
from those of longitude. If we suppose in the quadrant TT, 
points taken to mark the progress of the sun from day to day, and 
let meridians like PSE pass through these points, the arc of the 
ecliptic embraced between the meridians will be the daily incre- 
ments of longitude, while the corresponding parts of the equinoc- 
tial will be the daily increments of right ascension. Near T, the 
oblique direction in which the ecliptic cuts the meridian, will make 
the daily increments of longitude exceed those of right ascension; 
but this advantage is diminished as we approach the tropic, where 
the ecliptic becomes less oblique, and finally parallel to the equi- 
noctial ; while the convergence of the meridians contributes still 
farther to lessen the ratios of the daily increments of longitude to 
those of right ascension. Hence, at first, the diurnal arcs of 
right ascension are less than those of longitude, but afterwards 
greater ; and they continue greater for a similar distance beyond 

the tropic. 
^- 

108. From the foregoing considerations it appears, that the 
diurnal arcs of right ascension, by which the difference between 
the sidereal and the solar days is measured, are unequal, on ac- 
count both of the unequal motion of the sun in his orbit, and of 
the inclination of his orbit to the equinoctial. 

109. As astronomical time commences when the sun is on the 
meridian, so sidereal time commences when the vernal equinox 
is on the meridian, and is also counted from to 24 hours. By 
3 o'clock, for instance, of sidereal time, we mean that it is three 



THE CALENDAR. 45 

hours since the vernal equinox crossed the meridian ; as we say it 
is 3 o'clock of astronomical or of civil time, when it is three hours 
since the yun crossed the meridian. 



THE CALENDAR. 

110. The astronomical year is the time in which the sun makes 
one revolution in the ecliptic, and consists of 365d. 5h. 48m. 51 a .60. 
The civil year consists of 365 days. The difference is nearly 6 
hours, making one day in four years. 

111. The most ancient nations determined the number of days 
in the year by means of the stylus, a perpendicular rod which 
cast its shadow on a smooth plane, bearing a meridian line. The 
time when the shadow was shortest, would indicate the day of 
the summer solstice ; and the number of days which elapsed until 
the shadow returned to the same length again, would show the 
number of days in the year. This was found to be 365 whole 
days, and accordingly this period was adopted for the civil year. 
Such a difference, however, between the civil and astronomical 
years, at length threw all dates into confusion. For, if at first 
the summer solstice happened on the 21st of June, at the end of 
four years, the sun would not have reached the solstice until the 
22d of June, that is, it would have been behind its time. At the 
end of the next four years the solstice would fall on the 23d ; 
and in process of time it would fall successively on every day of 
the year. The same would be true of any other fixed date. 
Julius Caesar made the first correction of the calendar, by intro- 
ducing an intercalary day every fourth year, making February 
to consist of 29 instead of 28 days, and of course the whole year 
to consist of 366 days. This fourth year was denominated Bis- 
sextile.* It is also called Leap Year. 

I 

112. But the true correction was not 6 hours, but 5h. 49m.; 

hence the intercalation was too great by 1 1 minutes. This small 
fraction would amount in 100 years to of a day, and in 1000 

* The sextus dies ante Kalendas being reckoned twice, (Bis). 



46 THE EARTH. 

- 

years to more than 7 days. From the year 325 to 1582, it had 
in fact amounted to about 10 days; for it was known that in 325, 
the vernal equinox fell on the 21st of March, whereas, in 1582 it 
fell on the llth. In order to restore the equinox to the same date, 
Pope Gregory XIII decreed, that the year should be brought for- 
ward ten days, by reckoning the 5th of October the 15th. In or- 
der to prevent the calendar from falling into confusion afterwards, 
the following rule was adopted : 

Every year whose number is not divisible by 4 without a re- 
mainder, consists of 365 days ; every year which is so divisible, but 
is not divisible by 100, of 366 ; every year divisible by 100 but not 
by 400, again of 365 ; and every year divisible by 400, of 366. 

Thus the year 1838, not being divisible by four, contains 365 days, 
while 1836 and 1840 are leap years. Yet to make every fourth 
year consist of 366 days would increase it too much by about f 
of a day in 100 years; therefore every hundredth year has only 
365 days. Thus 1800, although divisible by 4, was not a leap 
year, but a common year. But we have allowed a whole day 
in a hundred years, whereas we ought to have allowed only three 
fourths of a day._ Hence, in 400 years we should allow a day too 
much, and therefore we let the 400th year remain a leap year. 
This rule involves an error of less than a day in 4237 years.* If 
the rule were extended by making every year divisible by 4,000 
(which would now consist of 366 days) to consist of 365 days, the 
error would not be more than one day in 100,000 years.f 

113. This reformation of the calendar was not adopted in Eng- 
land until 1752, by which time the error in the Julian calendar 
amounted to about 1 1 days. The year was brought forward, by 
reckoning the 3d of September the 14th. Previous to that time 
the year began the 25th of March ; but it was now made to be* 
gin on the 1st of January, thus shortening the preceding year, 
1751, one quarter.J 



* Wgodhouse, p. 874. t Herschel's Ast. p. 384. 

J Russia, and the Greek Church generally, adhere to the old style. In order to make 
the Russian dates correspond to ours, we must add to them 12 days. France and other 
Catholic countries, adopted the Gregorian calendar soon after it was promulgated. 



THE CALENDAR. 47 

114. As in the year 1582, the error in the Julian calendar 
amounted to 10 days, and increased by of a day in a ce*htury, 
at present the correction is 12 days ; and the number of the year 
will differ by one with respect to dates between the 1st of Janu- 
ary and the 25th of March. 

Examples. General Washington was born Feb. 11, 1731, old 
style ; to what date does this correspond in new style ? 

As the date is the earlier part of the 18th century, the correc- 
tion is 1 1 days, which makes the birth day fall on the 22d of 
February ; and since the year 1731 closed the 25th of March, 
wliile according to new style 1732 would have commenced on 
the preceding 1st of January ; therefore, the time required is Feb. 
22, 1732. It is usual, in such cases, to write b6th years, thus: 
Feb. 11, 1731-2, O. S. 

2. A great eclipse of the sun happened May 15th, 1836 ; to 
what date would this time correspond in old style ? 

Ans. May. 3d. 

115. The common year begins and ends on the same day of 
the week ; but leap year ends one day later in the week than it began. 

For 52x7=364 days; if therefore the year begins on Tues- 
day, for example, 364 days would complete 52 weeks, and one 
day would be left to begin another week, and the following year 
would begin on Wednesday. Hence, any day of the month is one 
day later in the week than the corresponding day of the preceding 
year. Thus, if the 16th of November, 1838, falls on Friday, 
the 16th of November, 1837, fell on Thursday, and will fall in 
1839 on Saturday. But if leap year begins on Sunday, it ends 
on Monday, and the following year begins on Tuesday ; while 
any given day of the month is two days later in the week than 
the corresponding date of the preceding year. 

116. Fortunately for astronomy, the confusion of dates involved 
in different calendars affects recorded observations but little. Re- 
markable eclipses, for example, can be calculated back for several 
thousand years, without any danger of mistaking the day of their 
occurrence ; and whenever any such eclipse is so interwoven with 
the account given by an ancient author of some historical event. 



48 THE EARTH. 

as to indicate precisely the interval of time between the eclipse 
and the event, and at the same time completely to identify the 
eclipse, that date is recovered and fixed forever.* 



CHAPTER V. 

OF ASTRONOMICAL INSTRUMENTS AND PROBLEMS FIGURE AND DEN- 
SITY OF THE EARTH. 

117. THE most ancient astronomers employed no instruments 
for measuring angles, but acquired their knowledge of the heav- 
enly bodies by long continued and most attentive inspection with 
the naked eye. In the Alexandrian school, about 300 years before 
the Christian era, instruments began to be freely used, and thence- 
forward trigonometry lent a powerful aid to the science of astron- 
omy. Tycho Brahe, in the 16th century, formed a new era in 
practical astronomy, and carried the measurement of angles to 
10", a degree of accuracy truly wonderful, considering that he 
had not the advantage of the telescope. By the application of 
the telescope to astronomical instruments, a far better defined view 
of objects was acquired, and a far greater degree of refinement 
was attainable. The astronomers royal of Great Britain perfected 
the art of observation, bringing the measurement of angles to 1", 
and the estimation of differences of time to T V of a second. Be- 
yond this degree of refinement it is supposed that we cannot 
advance, since unavoidable errors arising from the uncertainties 
of refraction, and the necessary imperfection of instruments, for- 
bid us to hope for a more accurate determination than this. But 
a little reflection will -show us, that I" on the limb of an astro- 
nomical instrument, must be a space exceedingly small. Suppose 
the circle, on v which the angle is measured, be one foot in diameter. 



* An elaborate view of the Calendar may be found in Delambre's Astronomy, t. III. 
A useful table for finding the day of the week of any given date, is inserted in the 
American Almanac for 1832, p. 72. 



ASTRONOMICAL INSTRUMENTS. 49 

Then 12X 3- 14159 = TV inch -space occupied .by 1. Hence 

ouO 

= space of 1', and =space of I". Such minute 



10x60 600 36000 

angles can be measured only by large circles. If. for example, 
a circle is 20 feet in diameter, a degree on its periphery would 
occupy a space 20 times as large as a degree on a circle of 1 foot. 
A degree therefore of the limb of such an instrument would 
occupy a space of 2 inches : one minute, ^ of an inch ; and one 
second, T VF of an inch. 

118. But the actual divisions on the limb of an astronomical 
instrument never extend to seconds : in the smaller instruments 
they reach only to 10', and on the largest rarely lower than 1'. 
The subdivision of these spaces is carried on by means of the 
Vernier, which may be thus defined : 

A VERNIER is a contrivancce attached to the graduated limb of 
an instrument, for the purpose of measuring aliquot parts of the 
smallest spaces, into which the instrument is divided. 

The vernier is usually a narrow zone of metal, which is made 
to slide on the graduated limb. Its divisions correspond to those 
on the limb, except that they are a little larger,* one tenth, for 
example, so that ten divisions on the vernier would equal eleven 
on the limb. Suppose now that our instrument is graduated to 
degrees only, but the altitude of a certain star is found to be 40 
and a fraction, or 40 +x. In order to estimate the amount of this 
fraction, we bring the zero point of the vernier to coincide with 
the point which indicates the exact altitude, or 40 +x. We then 
took along the vernier until we find where one of its divisions 
coincides with one of the divisions of the limb. Let this be at the 
fourth division of the vernier. In four divisions, therefore, the ver- 
nier has gained upon the divisions of the limb, a space equal to x ; 
and since, in the case supposed, it gains T V of a degree, or 6' at each 
division, the entire gain is 24', and the arc in question is 40 24'. 

119. As the vernier is used in the barometer, where its applica- 

* In the more modern instruments the divisions of the vernier are smaller than those 
of the limb, 

7 



THE EARTH. 




-31 



.GO 



-29 



B 



tion is more easily seen than in Economical instruments, while the 
principle is the same in both cases, let us Fig. 13. 

see how it is applied to measure the ex- 
act height of a column of mercury. Let 
AB (Fig. 13,) represent the upper part 
of a barometer, the level of the mercury 
being at C, namely, at 30.3 inches, and 
nearly another tenth. The vernier being 
brought (by a screw which is usually at- 
tached to it) to coincide with the surface 
of the mercury, we look along down the 
scale, until we find that the coincidence 
is at the 8th division of the vernier. 
Now as the vernier gains T V of i-V^T^o- 
of an inch at each division upward, it of 
course gains T F in eight divisions. The fractional quantity, there- 
fore, is .08 of an inch, and the height of the mercury is 30.38. If 
the divisions of the vernier were such, that each gained F V (when 
60 on the vernier would equal 61 on the limb) on a limb divided 
into degrees, we could at once take off minutes ; and were the limb 
graduated to minutes, we could in a similar way read off seconds. 

120. The instruments most used for astronomical observations, 
are the Transit Instrument, the Astronomical Clock, the Mural 
Circle, and the Sextant. A large portion of all the observations, 
made in an astronomical observatory, are taken on the meridian. 
When a heavenly body is on the meridian, being at its highest 
point above the horizon, it is then least affected by refraction and 
parallax ; its zenith distance (from which its altitude and decli- 
nation are easily derived) is readily estimated ; and its right as- 
cension may be very conveniently and accurately determined by 
means of the astronomical clock. Having the right ascension 
and declination of a heavenly body, various other particulars re- 
specting its position may be found, as we shall see hereafter, by 
the aid of spherical trigonometry. Let us then first turn our at- 
tention to the instruments employed for determining the right 
ascension and declination. They are the Transit Instrument, the 
Astronomical Clock, and the Mural Circle. 



ASTRONOMICAL INSTRUMENTS. 



51 



121. The Transit Instrument is a telescope, which is fixed 
permanently in the meridian, and moves only in that plane. It 
rests on a horizontal axis, which consists of two hollow cones 
applied base to base, a form uniting lightness and strength. The 
two ends of the axis rest on two firm supports, as pillars of stone, 
for example, usually built up from th ground, and so related to 
the building as to be as free as possible from all agitation. In 
figure 14, AD represents the telescope, E, W, massive stone pillars 
supporting the horizontal axis, beneath which is seen a spirit level, 
(which is used to bring the axis to a horizontal position,) and n a 
vertical circle graduated into degrees and minutes. This circle 
serves the purpose of placing the instrument at any required alti- 
tude or distance from the zenith, and of course for determining 
altitudes and zenith distances. 

Fig. 14. 




122. Various methods are described in works on practical as- 
tronomy, for placing the Transit Instrument accurately in the 
meridian. The following method by observations on the pole 
star, may serve as an example. If the instrument be directed 



52 



THE EARTH. 



towards the north star, and so adjusted that the star Alioth (the 
first in the tail of the Great Bear) and the pole star are both in 
the i same vertical circle, the former below the pole and the latter 
above it, the instrument will be nearly in the plane of the meridian. 
To adjust it more exactly, compare the time occupied by the pole 
star in passing from its upper to its lower culmination, with the 
time of passing from its lower to its upper culmination. These 
two intervals ought to be precisely equal ; and if they are so, the 
iustrument is truly placed in the meridian ; but if they are not 
equal, the position of the instrument must be shifted until they 
become exactly equal. 

123. The line of collimation of a telescope, is a line joining the 
center of the object glass with the center of the eye glass. When 
the transit instrument is properly adjusted, this line, as the instru- 
ment is turned on its axis, moves in the plane of the meridian. 
Having, by means of the vertical circle n, set the instrument at 
the known altitude or zenith distance of any star, upon which we 
wish to make observations, we wait until the star enters the field 
of the telescope, and note the exact instant when it crosses the 
vertical wire in the center of the field, which wire marks the true 
plane of the meridian. Usually, however, there are placed in the 
focus of the eye glass five parallel wires or threads, two on each 
side of the central wire, and all 
at equal distances from each 
other, as is represented in the 
following diagram. The time 
of arriving at each of the wires 
being noted, and all the times 
added together and divided by 
the number of observations, the 
result gives the instant of cross- 
ing the central wire. 



124. The Astronomical Clock 
is the constant companion of the 
Transit Instrument. This clock is so regulated as to keep exact 
pace with the stars, and of course with the revolution of the earth 




ASTRONOMICAL INSTRUMENTS. 53 

on its axis ; that is, it is regulated to sidereal time. It measures 
the progress of a star, indicating an hour for every 15, and 24 
hours for the whole period of the revolution of the star. Sidereal 
time, it will be recollected, commences when the vernal equinox 
is on the meridian, just as solar time commences when the sun is 
on the meridian. Hence, the hour by the sidereal clock has no 
correspondence with the hour of the day, but simply indicates 
how long it is since the equinoctial point crossed the meridian. 
For example, the clock of an observatory points to 3h. 20m. ; this 
may be in the morning, at noon, or any other time of the day, since 
it merely* shows that it is 3h. 20m. since the equinox was on the 
meridian. Hence, when a star is on the meridian, the clock 
itself shows its right ascension ; and the interval of time between 
the arrival of any two stars upon the meridian, is the measure of 
their difference of right ascension. 

125. Astronomical clocks are made of the best workmanship, 
with a compensation pendulum, and every other advantage which 
can promote their regularity. The Transit Instrument itself, 
when once accurately placed in the meridian, affords the means 
of testing the correctness of the clock, since one revolution of a 
star from the meridian to the meridian again, ought to correspond 
to exactly 24 hours by the clock, and to continue the same from 
day to day ; and the right ascension of various stars, as they cross 
the meridian, ought to be such by the clock as they are given in 
the tables, where they are stated according to the most accurate 
determinations of astronomers. Or by taking the difference of 
right ascension of any two stars on successive days, it will be seen 
whether the going of the clock is uniform for that part of the 
day ; and by taking the right ascension of different pairs of stars, 
we may learn the rate of the clock at various parts of the day. 
We thus learn, not only whether the clock accurately measures 
the length of the sidereal day, but also whether it goes uniformly 
from hour to hour. 

Although astronomical clocks have been brought to a great de- 
gree of perfection, so as to vary hardly a second for many months, 
yet none are absolutely perfect, and most are so far from it as to 
require to 'be corrected by means of the Transit Instrument every 



54 THE EARTH. 

few days. Indeed, for the nicest observations, it is usual not to 
attempt to bring the clock to an absolute state of correctness, but 
after bringing it as near to such a state as can conveniently be 
done, to ascertain how much it gains or loses in a day ; that is, to 
ascertain its rate of going, and to make allowance accordingly. 

126. The vertical circle (n, Fig. 14,) usually connected with 
the Transit Instrument, affords the means of measuring arcs on 
the meridian, as meridian altitudes, zenith distances, and decli- 
nations ; but l as the circle must necessarily be small, and there- 
fore incapable of measuring very minute angles, the Mural Cir- 
cle is usually employed for measuring arcs of the meridian. The 
Mural Circle is a graduated circle, usually of very large size, fixea 
permanently in the plane of the meridian, and attached firmly to 
a perpendicular wall. It is made of large size, sometimes 1 1 feet 
in diameter, in order that very small angles may be measured on 
its limb ; and it is attached to a massive wall of solid masonry in 
order to insure perfect steadiness, a point the more difficult to 
attain in proportion as the instrument is heavier. The annexed 
diagram represents a Mural Circle fixed to its wall and ready for 
observations. It will be seen that every expedient is employed 
to give the instrument firmness of parts and steadiness of position. 
Its radii are composed of hollow cones, uniting lightness and 
strength, and its telescope revolves on a large horizontal axis, 
fixed as firmly as possible in a solid wall. The graduations are 
made on the outer rim of the instrument, and are read off by six 
microscopes (called reading microscopes) attached to the wall, one 
of which is represented at A, and the places of the five others 
are marked by the letters B, C, D, E, F. Six are used, in order 
that by taking the mean of such a number of readings, a higher 
degree of accuracy may be insured, than could be obtained by a 
single reading. In a circle of six feet diameter, like that repre- 
sented in the figure, the divisions may be easily carried to five 
minutes each. The microscope (which is of the variety called 
compound microscope) forms an enlarged image of each of these 
divisions in the focus of the eye glass. With it is combined the 
principle of the micrometer. This is effected by placing in the 
focus a delicate wire, which may be moved by means of a screw 



ASTRONOMICAL INSTRUMENTS. 

Fig. 16. 



55 




in a direction parallel to the divisions of the limb, and which is so 
adjusted to the screw as to move over the whole magnified space 
of five minutes by five revolutions of the screw. Of course one 
revolution of the screw measures one minute. Moreover, if the 
screw itself is made to carry an index attached to its axis and re- 
volving with it over a disk graduated into sixty equal parts, then 
the space measured by moving the index over one of these parts, 
will be one second. 

We have been thus minute in the description of this instrument, 
in order to give the learner some idea of the vast labor and great 
patience demanded of practical astronomers, in order to obtain 
measurements of such extreme accuracy as those to which they 
aspire. 

On account of the great dimensions of this circle, and the ex- 
pense attending it, as well as the difficulty of supporting it firmly, 
sometimes only one fourth of it is employed, constituting the Mu- 
ral Quadrant. This instrument has the disadvantage, however, 



56 THE EARTH. 

of being applicable to only one hemisphere at a time, either the 
northern or the southern, according as it is fixed to the eastern 
or the western side of the wall." 

127. We have before shown (Art. 124,) the method of finding 
the right ascension of a star by means of the Transit Instrument 
and the clock. The declination may be obtained by means of the 
mural circle in several different ways, our object being always to 
find the distance of the star, when on the meridian, from the equa- 
tor (Art. 37.) First, the declination may be found from the me- 
ridian altitude. Let S (Fig. 17,) be the place of a star when 
on the meridian. Then its meridian altitude will be SH, which 
will best be found by taking its ze- 
nith distance ZS, of which it is the 

complement. From SH, subtract EH, 
the elevation of the equator, which 
equals the co-latitude of the place of 
observation, (Art. 44,) and the remain- 
der SE is the declination. Or if the 
star is nearer the horizon than the 
equator is, as at S', subtract its me- 
ridian altitude from the co-latitude, for "^ N" 
the declination. Secondly, the declination may be found from 
the north polar distance, of which it is the complement. Thus 
from P to E is 90. Therefore, PE-PS=90-PS=SE=the 
declination. The height of the pole P is always known when the 
latitude of the place is known, being equal to the latitude. 

128. The astronomical instruments already described are adapt- 
ed to taking observations on the meridian only ; but we some- 
times require to know the attitude of a celestial body when it is 
not on the meridian, and its azimuth, or distance from the meridian 
measured on the horizon ; and also the angular distance between 
two points on any part of the celestial sphere. An instrument 
especially designed to measure altitudes and azimuths, is called an 
Altitude and Azimuth Instrument, whatever may be its particular 
form. When a point is on the horizon its distance from the me- 
ridian, or its azimuth, may be taken by the common surveyor's 




ASTRONOMICAL INSTRUMENTS. 



57 



compass, the direction of the meridian being determined by the 
needle ; but when the object, as a star, is not on the horizon, its 
azimuth, it must be remembered, is the arc of the horizon from 
the meridian to a vertical circle passing through the star (Art. 27) ; 
at whatever different altitudes, therefore, two stars may be, and 
however the plane which passes through them may be inclined to 
the horizon, still it is their angular distance measured on the hori- 
zon which determines their difference of azimuth. Figure 18 rep- 
resents an Altitude and Azimuth Instrument, several of the, usual 
appendages and subordinate contrivances being omitted for the 
sake of distinctness and simplicity. Here abc is the vertical or 
altitude circle, and EFG the horizontal or azimuth circle ; AB is a 

Fig. 18. 




telescope mounted on a horizontal axis and capable of two mo- 
tions, one in altitude parallel to the circle abc, and the other in 
azimuth parallel to EFG. Hence it can be easily brought to bear 
upon any object. At m, under the eye glass of the telescope, is a 
small mirror placed at an angle of 45 with the axjs of the tele- 
scope, by means of which the image of the object is reflected up- 
wards, so as to be conveniently presented to the eye of the or> 

8 



58 THE EARTH. 

server. At d is represented a tangent screw! by which a slow 
motion is given to the telescope at c. At h and g are seen two 
spirit levels at right angles to each other, which show when the 
azimuth circle is truly horizontal. The instrument is supported 
on a tripod, for the sake of greater steadiness, each foot being 
furnished with a screw for levelling. 

129. The sextant is one of the most useful instruments, both 
to the astronomer and the navigator, and will therefore merit 
particular attention. In figure 19, 1 and H are two small mirrors, 
and T a small telescope. I D represents a movable arm, or 
radius, which carries an index at D. The radius turns on a pivot 
at I, and the index moves on a graduated arc EF. I is called 



19. 




the Index Glass and H the Horizon Glass. The under part only 
of the horizon glass is coated with quicksilver, the upper part 
being left transparent ; so that while one object is seen through 
the upper part by direct vision, another may be seen through 
the lower part by reflexion from the two mirrors. The instru- 
ment is so contrived, that when the index is moved up to F, 
where the zero point js placed, or the graduation begins, the two 



ASTRONOMICAL INSTRUMENTS. 59 

reflectors I and H are exactly parallel to each other. If we 
now look through the telescope, T, so pointed as to see the star 
S through the transparent part of the horizon glass, we shall 
see the same star, in the same place, reflected from the silvered 
part ; for the star (or any similar object) is at such a distance 
that the rays of light which strike upon the index glass I, are 
parallel to those which enter the eye directly, and will exhibit 
the object at the same place. Now, suppose we wish to meas- 
ure the angular distance between two bodies, as the moon and a 
star, and let the star be at S and the moon at M. The telescope 
being still directed to S, turn the index arm I D from P towards 
E until the image of the moon is brought down to S, its lower 
limb just touching S. By a principle in optics, the angular dis- 
tance which the image of the moon passes over, is twice that of 
the mirror I. But the mirror has passed over the graduated arc 
FD ; therefore double that arc is the angular distance between 
the star and the moon's lower limb. If we then bring the upper 
limb into contact with the star, the sum of both observations, 
divided by 2, will give the angular distance between the star 
and the moon's center. As each degree on the limb EF meas- 
ures two degrees of angular distance, hence the divisions for sin- 
gle degrees are in fact only half a degree asunder ; and a sextant, 
or the sixth part of the circle, measures an angular distance of 
120. The upper and lower points in the disk of the sun or of 
the moon, may be considered as two separate objects, whose 
distance from each other may be taken in a similar manner, 
and thus their apparent diameters at any time be determined. 
We may select our points of observation either in a vertical, or 
in a horizontal direction. 

130. If we make a star, or the limb of the sun or moon, one of 
the objects, and the point in the horizon directly beneath, the oth- 
er, we thus obtain the altitude of the object. In this observation, 
the horizon is viewed through the transparent part of the hori- 
zon glass. At sea, where the horizon is usually well defined, the 
horizon itself may be used for taking altitudes ; but on land, in- 
equalities of the earth's surface, oblige us to have recourse to an 
artificial horizon. This, in its simple state, is a basin of either 



60 THE EARTH. 

water or quicksilver. By this means we see the image of the 
sun (or other body) just as far below the horizon as it is in reality 
above it. Hence, if we turn the index glass until the limb of the 
sun, as reflected from it, is brought into contact with the image 
seen in the artificial horizon, we obtain double the altitude.* 

The sextant must be held in such a manner, that its plane shall 
pass through the plane of the two objects. It must be held 
therefore in a vertical plane in taking altitudes, and in -a horizontal 
plane in taking the horizontal diameters of the sun and moon. 
Holding the instrument in the true plane of the two bodies, whose 
angular distance is measured, is indeed the most difficult part of 
the operation. 

The peculiar value of the sextant consists in this, that the ob- 
servations taken with it are not affected by any motion in the 
observer ; hence it is the chief instrument used for angular meas- 
urements at sea. 

131. Examples illustrating the use of the Sextant. .. 
Ex. 1. Alt. O's lower limb, . . 49 10' 00" 
0's semi-diameter, . . 15 51 



49 25' 51" 
Subtract Refraction, . . 00 00 49 



49 25' 02" 
Add Parallax, . . . 00 00 06 



True altitude 0's center, . 49 25' 08" 

Ex. 2. With the Artificial Horizon. 

Altitude of 0's upper limb above the image in the artificial ho- 
rizon, 100 2' 47". 

True altitude, 50 01' 23."5 

0's serni-diameter, . . . 00 15 50. 

49 45' 33."5 
Refraction 00 00 48. 



49 44' 45."5 
Parallax . 00 00 05. 



True altitude of 0's center, . . . 49 44' 50."5 

* Woodhouse's Ast. p. 774. 




ASTRONOMICAL PROBLEMS. 61 

ASTRONOMICAL PROBLEMS.* 

132. Given the sun's Right Ascension and Declination, to find 
his Longitude and the Obliquity of the Ecliptic. 

Let PCP' (Fig. 20,) represent the celestial meridian that passes 
through the first of Cancer and Capricorn, (the solstitial colure,) 
PP' the axis of the sphere, EQ the equator, E'C the ecliptic, and 
PSP' the decimation circle (Art. 
37,) passing through the sun S ; . 
then ARS is a right angle, and in 
the right angled spherical triangle 
ARS, are given the right ascension 
AR (Art. 37,) and the declination 
RS, to find the longitude AS and 
the obliquity SAR. 

As longitude and right ascension 
are measured from A, the first point 

of Aries, in the direction AS of the signs, quite round the globe, 
when, of the four quantities mentioned in the problem, the obliquity 
and the declination are given to find the others, we must know 
whether the sun is north, or whether it is south of the equator, the 
longitude being in the one case AS, and in the other, instead of 
AS', it is 360 AS', that is, the supplement of AS'. We must 
also know on which side of the tropic the sun is, for the sun in 
passing from one of the tropics to the equinox, passes through the 
same degrees of declination as it had gone through in ascending 
from the other equinox to the tropic, although its longitude and 
right ascension go on continually increasing. From the 21st of 
March to the 21st of June, while describing the first quadrant 
from the vernal equinox, the declination is north and increasing ; 
north but decreasing, in the second quadrant, until the 23d of 
September ; south and increasing in the third quadrant, until the 
21st of December; and finally, in the fourth quadrant, south but 
decreasing until the 21st of March. 

Ex. 1. On the 17th of May, the sun's Right Ascension was 
53 38', and his Decimation 19 15' 57": required his Longitude 
and the Obliquity of the Ecliptic. 

* Young's Spherical Trigonometry, p. 136. Vince's Complete System, Vol. I. 



62 THE EARTH. 

Applying Napier's rule* to the right angled triangle, ARS, we 
have 

1. Rad. cos. AS=cos. AR cos. RS. 

> T i A r A A rad. sin. AR 

2. Rad. sm. AR=tan. RS cot. A.*. cot. A= =-^ 

tan. RS 

Hence the computation for AS and A is as follows : 

For the Longitude AS. For the Obliquity A. 



cos.AR 53 38' 00" 9.7730185 
cos.RS 19 15 57 9.9749710 



cos.AS 55 57 43 9.7479895 



sm AR 9.9059247 

tan. RS, ar. com. 0.4565209 



cot. A 23 27' 50$" 10.3624456 



Ex. 2. On the 31st of March, 1816, the sun's Declination was 
observed at Greenwich to be 4 13' 31^": required his Right 
Ascension, the obliquity of the ecliptic being 23 27' 51". 

Ans. 9 47' 59". 

Ex. 3. What was the sun's Longitude on the 28th of Novem- 

* The student *s supposed to be acquainted with Spherical Trigonometry ; but to re- 
fresh his memory, we may insert a remark or two. 

It will be recollected that in Napier's rule for the solution of a right angled spherical 
triangle, by means of the Five Circular Parts, we proceed as follows. 

In a right angled spherical triangle we are to recognize but five parts, viz. the three 
sides and the two oblique angles. If we take any one of these as a middle part, the 
two which lie next to it on each side will be adjacent parts. Thus, (in Fig. 21,) taking 
A for a middle part, b and c will be the adjacent parts ; if we take c for the middle part, 
A and B will be the adjacent parts ; if we fig. 21. 

take B for the middle part, c and a will be 
the adjacent parts ; but if we take a for 
the middle part, then as the angle C is 
not considered as one of the circular parts, 
B and b are the adjacent parts ; and, last- 
ly, if b is the middle part, then the adja- 
cent parts are A and a. The two parts immediately beyond the adjacent parts on each 
side, still disregarding the right angle, are called the opposite parts ; thus if A is the 
middle part, the opposite parts are a and B. Napier's rule is as follows : 

Radius into the sine of the middle part, equals the product of the tangents of the 
adjacent extremes, or of the cosines of the opposite extremes. 

(The corresponding vowels are marked to aid the memory.) This rule is modified 
by using the complements of the two angles and the hypothenuse instead of the parts 
themselves. Thus instead of rad.Xsin. A, we say rad.Xcos. A, when A is the middle 
part ; and rad.Xcos. AB, when the hypothenuse is the middle part. 

Examples. 1. In the right angled triangle ABC, are given the two perpendicular 
sides, viz. a 48 24' 10", 6=59 38' 27", to find the hypothenuse c. The hypothenuse 
being made the middle part, the other sides become the opposite parts, being separated 




ASTRONOMICAL PROBLEMS. 63 

ber, 1810, when his Declination was 21 16' 4", and his Right 
Ascension, in time, 16h. 14m. 58.4s.? 

Ans. 245 39' 10". 

Ex. 4. The sun's Longitude being 8s. 7 40' 56", and the Ob- 
liquity 23 27' 42|", what was the Right Ascension in time? 

Ans. 16h. 23m. 34s. 

133. Given the sun's Declination to find the time of his Rising 
and Setting at any place whose latitude is known. 

Let PEP' (Fig. 22,) represent the meridian of the place, Z 
being the zenith, and HO the A horizon ; and let LL' be the appa- 
rent path of the sun on the proposed 
day, cutting the horizon in S. Then 
the arc EZ will be the latitude of the 
place, and consequently EH, or its 
equal QO, will be the co-latitude, and 
this measures the angle OAQ ; also 
RS will be the sun's declination, and 
AR expressed in time will be the time 
of rising before 6 o'clock. For it is 
evident that it will be sunrise when 

the sun arrives at the horizon at S ; but PP' being an hour circle 
whose plane is perpendicular to the meridian, (and of course pro- 
jected into a straight line on the plane of projection,) the time the 
sun is passing from S to S' taken from the time of describing S'L, 
which is six hours, must be the time from midnight to sunrise. 
But the time of describing SS' is measured on the corresponding 
arc of the equinoctial AR. 

In the right angled triangle ARS, we have the declination RS, 
and the angle A to find AR. Therefore, 
Rad. xsin. AR=cot. A xtan. RS. 

from the middle part by the angles A and B. Hence, rad. cos. c=cos. a cos. 6 .*. cos* c= 
coB.aeoB.fe ' 

rad. 

2. In the spherical triangle, right angled at C, are given two perpendicular sides, 
viz. a=116o 30' 43", 4=29 41' 32", to find the angle A. 

Here, the required angle is adjacent to one of the given parts, viz. 6, which make 
the middle part. Then, 

Rad.xsin. 6=cot A tan. a .-.cot. A= rad ' Xsin ' 6 =76Q 7' 13*. 

tan. a. 




64 



THE EARTH. 



Ex. 1. Required the time of sunrise at latitude 52 13' N. 
when the sun's declination is 23 28'. 

Rad 10. 

Cot. A or tan. 52 13' 10.1105786 

Tan. RS= 23 28' 9.6376106 

Sin. 34 03 211" } 



4* 



9.7481892 



2h. 16' 13" 25"' ) 

6 

3h. 43' 46" 35'"= the time after midnight, and of 
course the time of rising. 

Ex. 2. Required the time of sunrise at latitude 57 2' 54" N. 
when the sun's declination is 23 28' N. 

Ans. 3h. llm. 49s. 

Ex. 3. How long is the sun above the horizon in latitude 58 
12' N. when his declination is 18 40' S. ? 

Ans. 7h. 35m. 52s. 



Fig. 23. 



134. Given the Latitude of the place, and the Declination of a 
heavenly body, to determine its Altitude and Azimuth when on the 
six o 'clock hour circle. 

Let HZO (Fig. 23.) be the meridian of the place, Z the zenith 
HO the horizon, S the place of 
the object on the 6 o'clock hour 
circle PSP', which of course cuts 
the equator in the east and west 
points, and ZSB the vertical cir- 
cle passing through the body. 
Then in the right angled triangle 
SBA, the given quantities are 
AS, which is the declination, 
and the arc OP or angle SAB, 
the latitude of the place, to find 
the altitude BS, and the azimuth 
BO, or the amplitude AB, which is its complement. 

Ex. 1. What were the altitude and azimuth of Arcturus, wfien 




Degrees are converted into hours by multiplying by 4 and dividing by 60. 



ASTRONOMICAL PROBLEMS. 



65 



upon the six o'clock hour circle of Greenwich, lat: 51 28' 40" N. 
on the first of April, 1822 ; its declination being 20 6' 50" N. ? 



For the Altitude. 

Rad. sin. BS=sin. AS sin. A 
Rad. . . 10. 
Sin. 20 06' 50" 9.5364162 
Sin. 51 28 40 9.8934103 


For the Azimuth. 

Rad. cos. A=cot. BO cot. AS 
Cot. 20 06' 50" 10.4362545 
Cos. 51 28 40 9.7943612 
Rad. . . 10. 


Sin. 15 36 27 9.4298265 


Cot. 77 09' 04" 9.3581067 




Ex. 2. At latitude 62 12' N. the altitude of the sun at 6 o'clock 
in the morning was found to be 18 20' 23": required his declina- 
tion and azimuth. 

Ans. Dec. 20 50' 12" N. Az. 79 56' 4". 

135. The Latitudes and Longitudes of two celestial objects be- 
ing given, to find their Distance apart. 

Let P (Fig. 24,) represent the pole of the ecliptic, and PS, PS', 
two arcs of celestial latitude (Art. 37,) drawn to the two objects 
SS' ; then will these arcs represent the Fig. 24 

co-latitudes, the angle P will be the 
difference of longitude, and the arc SS' 
will be the distance sought. Here we 
have the two sides and the included 
angle given to find the third side. By 
Napier's Rules for the solution of oblique angled spherical triangles, 
(see Spherical Trigonometry,) the sum and difference of the two 
angles opposite the given sides may be found, and thence the an- 
gles themselves. The required side may then be found by the theo- 
rem, that the sines of the sides are as the sines of their opposite 
angles.* The computation is omitted here on account of its great 
length. If P be the pole of the equator instead of the ecliptic, 
then PS and PS' will represent arcs of co-declination, and the 
angle P will denote difference of right ascension. From these 
data, also, we may therefore derive the distance between any two 
stars. Or, finally, if P be the pole of the horizon, the angle at P 

* More concise formulae for the solution of this case may be found in Young's Tri- 
gonometry, p. 99. Francceur's Uranography, Art. 330. Dr. Bowditch's Practical 
Navigator, p, 436. 

9 



66 THE EARTH. 

will denote difference of azimuth, and the sides PS, PS', zenith 
distances, from which the side SS' may likewise be determined. 

FIGURE AND DENSITY OF THE EARTH. 

136. We have already shown, (Art. 8,) that the figure of the 
earth is nearly globular ; but since the semi-diameter of the earth 
is taken as the base line in determining the parallax of the heav- 
enly bodies, and lies therefore at the foundation of all astronomi- 
cal measurements, it is very important that it should be ascertained 
with the greatest possible exactness. Having now learned the 
use of astronomical instruments, and the method of measuring 
arcs on the celestial sphere, we are prepared to understand the 
methods employed to determine the exact figure of the earth. 
This element is indeed ascertained in four different ways, each 
of which is independent of all the rest, namely, by investigating 
the effects of the centrifugal force arising from the revolution of 
the earth on its axis by measuring arcs of the meridian by 
experiments with the pendulum and by the unequal action of the 
earth on the moon, arising from the redundance of matter about 
the equatorial regions. We will briefly consider each of these 
methods. 

137. First, the known effects of the centrifugal force, would give 
to the earth a spheroidal figure, elevated in the equatorial, and flat 
tened in the polar regions. 

Had the earth been originally constituted (as geologists sup 
pose) of yielding materials, either fluid or semi-fluid, so that 
its particles could obey their mutual attraction, while the body 
remained at rest it would spontaneously assume the figure of a 
perfect sphere ; as soon, however, as it began to revolve on its 
axis, the greater velocity of the equatorial regions would give to 
them a greater centrifugal force, and cause the body to swell out 
into the form of an oblate spheroid.* Even had the solid part of 
the earth consisted of unyielding materials and been created a 
perfect sphere, still the waters that covered it would have receded 
from the polar and have been accumulated in the equatorial re- 

* See a good explanation of this subject in the Edinburgh Encyclopaedia, II. 665. 



FIGURE OF THE EARTH. 67 

gions, leaving bare extensive regions on the one side, and ascend- 
ing to a mountainous elevation on the other. 

On estimating from the known dimensions of the earth and 
the velocity of its rotation, the amount of the centrifugal force in 
different latitudes, and the figure of equilibrium which would 
result, Newton inferred that the earth must have the form of an 
oblate spheroid before the fact had been established by observa- 
tion ; andlie assigned neatly the true ratio of the polar and equa- 
torial diameters. 

138. Secondly, the spheroidal figure of the earth ^s proved, by 
actually measuring the length of a degree on the meridian in differ- 
ent latitudes. 

Were the earth a perfect sphere, the section of it made by a 
plane passing through its center in any direction would be a per- 
fect circle, whose curvature would be equal in all parts ; but if 
we find by actual observation, that the curvature of the section is 
not uniform, we infer a corresponding departure in the earth from 
the figure of a perfect sphere. This task of measuring portions of 
the meridian, has been executed in different countries by means 
of a system of triangles with astonishing accuracy.* The result 
is, that the length of a degree increases as we proceed from the 
equator towards the pole, as may be seen from the following table : 



Places of observation. 1 ' Latitude. 


Length of a degree in miles. 


Peru, 
Pennsylvania, 
Italy, 
France, 
England, 
Sweden, 


00 00' 00" 
39 12 00 
43 01 00 
46 12 00 
51 29 54| 
66 20 10 


68.732 
68.896 
68.998 
69.054 
69.146 
69.292 



Combining the results of various measurements, the dimensions 
of the terrestrial spheroid are found to be as follows : f 

Equatorial diameter, . . . 7925.308 

Polar diameter, .... 7898.952 

Mean diameter, . . . . 7912.130 

The difference between the greatest and least, is 26.356=^, 

* See Day's Trigonometry. t Beseel. 



THE EARTH. 



of the greatest. This fraction ( I T ) is denominated the elliplicity 
of the earth, being the excess of the transverse over the conjugate 
axis, on the supposition that the section of the earth coinciding 
with the meridian, is an ellipse : and that such is the case, is 
proved by the fact that calculations on this hypothesis, of the 
lengths of arcs of the meridian in different latitudes, agree nearly 
with the lengths obtained by actual measurement. 

139. Thirdly, the figure of the earth is shown to be spheroidal, by 
observations with the pendulum. 

The use of the pendulum in determining the figure of the 
earth, is founded upon the principle that the number of vibra- 
tions performed by the same pendulum, when acted on by differ- 
ent forces, varies as the square root of the forces.* Hence, by 
carrying a pendulum to different parts of the earth, and counting 
the number of vibrations it performs in a given time, we obtain 
the relative forces of gravity at those places, and this leads to a 
knowledge of the relative distance of each place from the center 
of the earth, and finally, to the ratio between the equatorial and 
the polar diameters. 

A 

140. Fourthly, that the earth is of a spheroidal figure, is infer- 

red from the motions of the moon. 

These are found to be affected by the excess of matter about 
the equatorial regions, producing certain irregularities in the lunar 
motions, the amount of which becomes a measure of the excess 
itself, and hence affords the means of determining the earth's 
ellipticity. This calculation has been made by the most profound 
mathematicians, and the figure deduced from this source corres- 
ponds very nearly to that derived from the several other indepen- 
dent methods. 

We thus have the shape of the earth established upon the most 
satisfactory evidence, and are furnished with a starting point from 
which to determine various measurements among the heavenly 
bodies. 

141. The density of the earth compared with water, that is, its 

* Mechanics, Art. 183. 



DENSITY OP THE EARTH. 



69 



specific gravity, is 5J.* The density was first estimated by Dr. 
Hutton, from observations made by Dr. Maskelyne, Astronomer 
Royal, on Schehallien, a mountain of Scotland, in the year 1774. 
Thus, let M (Fig. 25,) represent 
the mountain, D, B, two stations 
on opposite sides of the moun- 
tain, and I a star; and let IE 
and IG be the zenith distances as 
determined by the differences of 
latitudes of the two stations. But 
the apparent zenith distances as 
determined by the plumb line 
are IE' and IG'. The deviation 
towaras the mountain on each 
side exceeded 7".f The attrac- 
tion of the mountain being ob- 
served on both sides of it, and 
its mass being computed from a number of sections taken in all 
directions, these data, when compared with the known attraction 
and magnitude of the earth, led to a knowledge of its mean den- 
sity. .According to Dr. Hutton, this is to that of water as 9 to 2 ; 
but later and more accurate estimates have made the specific 
gravity of the earth as stated above. But this density is nearly 
double the average density of the materials that compose the ex- 
terior crust of the earth, showing a great increase of density 
towards the center. 

The density of the earth is an important element, as we shall 
find that it helps us to a knowledge of the density of each of the 
other members of the solar system. 




* Baily, Ast. Tables, p. 21. 



t Robison's Phys. Ast. 



PART II. OF THE SOLAR SYSTEM. 



142. HAVING considered the Earth, in its astronomical relations, 
and the Doctrine of the Sphere, we proceed now to a survey of 
the Solar System, and shall treat successively of the Sun, Moon, 
Planets, and Comets. 



CHAPTER I. 

OF THE SUN SOLAR SPOTS ZODIACAL LIGHT. 

143. THE figure which the sun presents to us is that of a per- 
fect circle, whereas most of the planets exhibit a disk mor*e or less 
elliptical, indicating that the true shape of the body is an oblate 
spheroid. So great, however, is the distance of the sun, that a 
line 400 miles long would subtend an angle of only I" at the eye, 
and would therefore be the least space that could be measured. 
Hence, were the difference between two conjugate diameters of 
the sun any quantity less than this, we could not determine by 
actual measurement that it existed at all. Still we learn from 
theoretical considerations, founded upon the known effects of cen- 
trifugal force, arising from the sun's revolution on his axis, that 
his figure is not a perfect sphere, but is slightly spheroidal.* 

144. The distance of the sun from the earth, is nearly 95,000,000 
miles. For, its horizontal parallax being 8."6, (Art. 86,) and the 
semi-diameter of the earth 3956 miles, 

Sin. 8."6 : 3956 : : Rad. : 95,000,000 nearly. In order to form 
some faint conception at least of this vast distance, let us reflect 
that a railway car, moving at the rate of 20 miles per hour, would 
require more than 500 years to reach the sun. 

* See Mecanique Celeste, III, 165. Delambre, 1. 1, p. 483. 



SOLAR SPOTS. 71 

145. The apparent diameter of the sun may be found either by 
^ the Sextant, (Art. 129,) by an instrument called the Heliometer, 
S specially designed for measuring its angular breadth, or by the time 

it occupies in crossing the meridian. If, for example, it occupied 
4 m , its angular diameter would be 1. It in fact occupies a little 
more than 2 m , and hence its apparent diameter is a little more than 
half a degree, (32' 3"). Having the distance and angular diameter, 
J/ ' we can easily find its linear diameter. Let E (Fig. 26,) be the 
earth, S the sun, ES a line drawn to the Fig. 26. 

center of the disk, and EC a line drawn 
touching the disk at C. Join SC ; then 

Rad. : ES (95,000,000) : : sin. 16' l."5 : 
442840=semi-diameter, and 885680=diam- 

eter. And =112 nearly ; that is, it 

would require one hundred and twelve bo- 
dies like the earth, if laid side by side, to 
reach across the diameter of the sun ; and a 
ship sailing at the rate of ten knots an hour, 
would require more than ten years to sail 
across the solar disk. Since spheres are to 
each other as the cubes of their diameters, 

I 3 : H2 3 :: 1 : 1,400,000 nearly; that is, the sun is about 
1,400,000 times as large as tft earth. The distance of the moon 
from the earth being 237,000 miles, were the center of the sun 
made to coincide with the center of the earth, the sun would ex- 
tend every way from the earth nearly twice as far as the moon. 

146. In density, the sun is only one fourth that of the earth, 
being but a little heavier than water (Art.. 141) ; and since the 
quantity of matter, or mass of a body, is proportioned to its mag- 
nitude and density, hence, 1,400,000 xj = 350,000, that is, the 
quantity of matter in the sun is three hundred and fifty thousand 
(or, more accurately, 354,936) times as great as in the earth. Now 
the weight of bodies (which is a measure of the force of gravity) 
varies directly as the quantity of matter, and inversely as the 
square of the distance. A body, therefore, would weigh 350,000 
times as much on the surface of the sun as on the earth, if the 




72 . THE SUN. 

distance of the center of force were the same in both cases ; but 
since the attraction of a sphere is the same as though all the mat- 
ter were collected in the center, consequently, the weight of a 
body, so far as it depends on its distance from the center of force, 
would be the square of 112 times less at the sun than at the earth. 
Or, putting W for the weight at the earth, and W 7 for the weight 
at the sun, then 



Hence a body would weigh nearly 28 times as much at the sun 
as at the earth. A man weighing 200 Ibs. would, if transported 
to the surface of the sun, weigh 5,580 Ibs., or nearly 2^ tons. To 
lift one's limbs, would, in such a case, be beyond the ordinary 
power of the muscles. At the surface of the earth, a body falls 
through 16 j % feet in a second ; and since the spaces are as the 
velocities, the times being equal, and the velocities as the forces, 
therefore a body would fall at the sun in one second, through 
16 T V X 27 T 9 o = 448.7 feet. 

SOLAR SPOTS. 

147. The surface of the sun, when viewed with a telescope, 
usually exhibits dark spots, which vary much, at different tim'es, 
in number, figure, and extent. One hundred or more, assembled 
in several distinct groups, are sometimes visible at once on the 
solar disk. The solar spots are commonly very small, but 
occasionally a spot of enormous size is seen occupying an extent 
of 50,000 miles or more in dianueter. They are sometimes 
1 even visible to the naked eye, when the sun is viewed through 
colored glass, or when near the horizon, it is seen through light 
clouds or vapors. When it is recollected that 1" of the solar 
disk implies an extent of 400 miles, (Art. 143,) it is evident that a 
space large enough to be seen by the naked eye, must cover a very 
large extent. 

A solar spot usually consists of two parts, the nucleus and the 
umbra, (Fig. 27.) The nucleus is black, of a very irregular shape, 
and is subject to great and sudden changes, both in form and size. 
Spots have sometimes seemed to burst asunder, and to project frag- 
ments in different directions. The umbra is a wide margin of lighter 



SOLAR SPOTS. 




shade, and is commonly of greater Fig. 27. 

extent than the nucleus. The spots 
are usually confined to a zone ex- 
tending -across the central regions 
of the sun, not exceeding 60 in 
breadth. When the spots are ob- 
served from day to day, they are 
seen to move across the disk of the 
sun, occupying about two weeks in 
passing from one limb to the other. 
After an absence of about the same 
period, the spot returns, having taken 27d. 7h. 37m. in the entire 
revolution. 

148. The spots must be nearly or quite in contact with the body 
of the sun. Were they at any considerable distance from it, the 
time during which they would be seen on the solar disk, would 
be .less than that occupied in the remainder of the revolution. 
Thus, let S (Fig. 28,) be the sun, E the earth, and abc the path 
of the body, revolving about the sun. 
Unless the spot were nearly or quite 
in contact with the body ff the sun, 
being projected upon his disk only 
while passing from b to c, and being 
invisible while describing the arc cab, 
it would of course be out of sight lon- 
ger, than in sight, whereas the two pe- 
riods are found to be equal. Moreover, 
the lines which all the solar spots de- 
scribe on the disk of the sun, are found 
to be parallel to each other, like the 
circles of diurnal revolution around the 
earth ; and hence it is inferred that 
they arise from a similar cause, namely, 
the revolution of the sun on his axis, 
a fact which is thus made known to 
us. 

But although the spots occupy <about 27 days in passing from 

10 




74 



THE SUN. 




one limb of the sun around to the same limb again, yet this is not 

the period of the sun's revolution on his axis, but exceeds it by 

nearly tw^> days. For, let AA'B (Fig. 29,) represent the sun, and 

EE'M the orbit of the earth. When the earth is at E, the 

visible disk of the sun will be AA'B ; 

and if the earth remained stationary at 

E, the time occupied by a spot after 

leaving A until it returned to A, would 

be just equal to the time of the sun's 

revolution on his axis. But during the 

27^ days in which the spot has been 

performing its apparent revolution, the 

earth has been advancing in his orbit 

from E to E', where the visble disk of 

the sun is A'B'. Consequently, before 

the spot can appear again on the limb from which it set out, it 

must describe so much more than an entire revolution as equals 

the arc AA ; , which equals the arc EE'. Hence, 

365d. 5h. 48m.+27d. 7h. 37m. : 365d. 5h. 48m. : : 27d. Jh. 37m. : 
25d. 9h. 59m.=the time of the sun's revolution on his axis. 

149. If the path which the spots Appear to describe by the re- 
volution of the sun on his axis left each a visible trace on his sur- 
face, they would form, like the circles of diurnal revolution on the 
earth, so many parallel rings, of which that which passed through 
the center would constitute the solar equator, while those on each 
side of this great circle would be small circles, corresponding to 
parallels of latitude on the earth. Let us conceive of an artifi- 
cial sphere to represent the sun, having such rings plainly marked 
on its surface. Let this sphere -be placed at some distance from 
the eye, with its axis perpendicular to the axis of vision, in which 
case the equator would coincide with the line of vision, and its 
edge be presented to the eye. It would therefore be projected in- 
to a straight line. The same would be the case with all the small- 
er rings, the distance being supposed such that the rays of light 
come from them all to the eye nearly parallel. Now let the axis, 
instead of being perpendicular to the line of vision, be inclined to 
that line, then all the rings being seen obliquely would be projected 



SOLAR SPOTS. 75 

into ellipses. If, however, while the sphere remained in a fixed 
position, the eye were carried around it, (being always in the same 
plane,) twice during the circuit it would be in the plane of the 
equator, and project this and all the smaller circles into straight 
lines ; and twice, at points 90 distant from the foregoing posi- 
tions, the eye would be at a distance from the planes of the rings 
equal to the inclination of the equator of the sphere to the line of 
vision. Here it would project the rings into wider ellipses than 
at other points ; and the ellipses would become more and more 
acute as the eye departed from either of these points, until they 
vanished again into straight lines. 

150. It is in a similar manner that the eye views the paths de- 
scribed by the spots on the sun. If the sun revolved on an axis 
perpendicular to the plane of the earth's orbit, the eye being situ- 
ated in the plane of revolution, and at such a distance from the 
sun that the light comes to the eye from all parts of the solar 
disk nearly parallel, the paths described by the spots would be 
projected into straight lines, and each would describe a straight 
line across the solar disk, parallel to the plane of revolution. But 
the axis of the sun is inclined to the ecliptic about 7j from a per 
pendicular, so that usually all the circles described by the spots are 
projected into ellipses. The breadth of these, however, will vary 
as the eye, in the annual revolution, is carried around the sun, and 
when the eye comes into the plane of the rings, as it does twice a 
year, they are projected into straight lines, and for a short time a 
spot seems moving in a straight line inclined to the plane of the 
ecliptic 7. The two points where the sun's equator cuts the 
ecliptic are called the sun's nodes. The longitudes of the nodes 
are 80 7' and 260 7', and the earth passes through them about 
the 12th of December, and the llth of June. It is at these times 
that the spots appear to describe straight lines. We have men- 
tioned the various changes in the apparent paths of the solar spots, 
which arise from the inclination of the sun's axis to the plane of 
the ecliptic ; but it was in fact by first observing these changes, 
and proceeding in the reverse order from that which we have pur- 
sued, that astronomers ascertained that the sun revolves on his 
axis, and that this axis is inclined to the ecliptie 82. 



76 



THE SUN. 



151. With regard to the cause of the solar spots, various hypo- 
theses have been proposed, none of which is entirely satisfactory. 
That which ascribes their origin to volcanic action, appears to us 
the most reasonable.* 

Besides the dark spots on the sun, there are also seen, in dif- 
ferent parts, places that are brighter than the neighboring por- 
tions of the disk. These are called faculce. Other inequalities 
are observable in powerful telescopes, all indicating that the sur- 
face of the sun is in a state of constant and powerful agitation. 

ZODIACAL LIGHT. 

152. The Zodiacal Light is a faint light resembling the tail of 
a c.omet, and is seen at certain seasons of the year following the 
course of the sun after evening twilight, or preceding his approach 
in the morning sky. Figure 30 represents its appearance as seen 
in the evening in March, 1836. The following are the leading 
facts respecting it. 

1 . Its form is that of a luminous Fiar. 30. 
pyramid, having its base towards 

the sun. It reaches to an immense 
distance from the sun, sometimes 
even beyond the orbit of the earth. 
It is brighter in the parts nearer the 
sun than in those that are more 
remote, and terminates in an ob- 
tuse apex, its light fading away by 
insensible gradations, until it be- 
comes too feeble for distinct vision. 
Hence its limits are, at the same 
time, fixed at different distances 
from the sun by different observers, 
according to their respective powers 
of vision. 

2. Its aspects vary very much with the different seasons of the 
year. About the firs* of October, in our climate, (Lat. 41 18',) 

* In the system of instruction in Yale College, subjects of this kind are discussed 
in a course of astronomical lectures, addressed to the class after they have finished the 
perusal of the text-book. 




ZODIACAL LIGHT. 77 

it becomes visible before the dawn of day, rising along north of 
the ecliptic, and terminating above the nebula of Cancer. About 
the middle of November, its vertex is in the constellation Leo. 
At this time no traces of it are seen in the west after sunset, but 
about the first of December it becomes faintly visible in the west, 
crossing the Milky Way near the horizon, and reaching from the 
sun to the head of Capricornus, forming, as its brightness increases, 
a counterpart to the Milky Way, between which on the right, 
and the Zodiacal Light on the left, lies a triangular space embra- 
cing the Dolphin. Through the month of December, the Zodi- 
acal Light is seen on both sides of the sun, namely, before the 
morning and after the evening twilight, sometimes extending 50 
westward, and 70 eastward of the sun at the same time. After 
it begins to' appear in the western sky, it increases rapidly from 
night to ight, both in length and brightness, and withdraws itself 
from the morning sky, where it is scarcely seen after the month 
of December, until the next October. 

3. The Zodiacal Light moves through the heavens in the order of 
the signs. It moves with unequal velocity, being sometimes sta- 
tionary and sometimes retrograde, while at other times it ad- 
vances much faster than the sun. In February and March, it is 
very conspicuous in the west, reaching to the Pleiades and be- 
yond ; but in April it becomes more faint, and nearly or quite dis- 
appears during the month of May. It is scarcely seen in this lat- 
itude during the summer months. 

4. It is remarkably conspicuous at certain periods of a few 
years, and then for a long interval almost disappears. 

5. The Zodiacal Light was formerly lield to be the atmosphere of 
the sun.* But La Place has shown that the solar atmosphere 
could never reach so far from the sun as this light is seen to ex- 
tend.-]- It has been supposed by others to be a nebulous body 
revolving around the sun. The idea has been suggested, that the 
extraordinary Meteoric Showers, which at different periods visit 
the earth, especially in the month of November, may be derived 
from this body.J 

* Mairan*Memoirs French Academy, for 1733. t Mec. Celeste, III, 525. 

i See note on " Meteoric Showers," at the end of the volume. 



CHAPTER II. 

OF THE APPARENT ANNUAL MOTION OF THE SUN SEASONS FIGURE 



153. THE revolution of the earth around the sun once a year, 
produces an apparent motion of the sun around the earth in the 
same period. When bodies are at such a distance from each 
other as the earth and the sun, a spectator on either would pro- 
ject the other body upon the concave sphere of the heavens, al- 
ways seeing it on the opposite side of a great circle, 180 from 
himself. Thus when the earth arrives at Libra (Fig. IT,) we see 
the sun in the opposite sign Aries. When the earth moves from 
Libra to Scorpio, as we are unconscious of our own motion, the 
sun it is that appears to move from Aries to Taurus^ being always 
seen in the heavens, where a line drawn from the eye of the spec- 
tator through the body meets the concave sphere of the heavens. 
Hence the line of projection carries the sun forward on one side 
of the ecliptic, at the same rate as the earth moves on the oppo- 
site side ; and therefore, although we are unconscious of our own 
motion, we can read it from day to day in the motions of the sun. 
If we could see the stars at the same time with the sun, we could 
actually observe from day to day the sun's progress through them, 
as we observe the progress of the moon at night ; only the sun's 
rate of motion would be nearly fourteen times slower than that 
of the moon. Although we do not see the stars when the sun is 
present, yet after the sun is set, we can observe that it makes daily 
progress eastward, as is apparent from the constellations of the 
Zodiac occupying, successively, the western sky after sunset, 
proving that either all the stars have a common motion westward 
independent of their diurnal motion, or that the sun has a motion 
past them, from west to east. We shall see hereafter abundant 
evidence to prove, that this change in the relative position of the 
sun and stars, is owing to a change in the apparent place of the 
sun, and not to any change in the stars. 



ANNUAL MOTION. 79 

154. Although the apparent revolution of the sun is in a direc- 
tion opposite to the real motion of the earth, as regards absolute 
space, yet both are nevertheless from west to east, since these 
terms do not refer to any directions in absolute space, but to the 
order in which certain constellations (the constellations of the 
Zodiac) succeed one another. The earth itself, on opposite sides 
of its orbit, does in fact move towards directly opposite points of 
space ; but it is all the while pursuing its course in the order of 
the signs. In the same manner, although the earth turns on its 
axis from west to east, yet any place on the surface of the earth 
is moving in a direction in space exactly opposite to its direction 
twelve hours before. If the sun left a visible trace on the face 
of the sky, the ecliptic would of course be distinctly marked on 
the celestial sphere as it is on an artificial globe ; and were the 
equator delineated in a similar manner, (by any method like that 
supposed in Art. 46,) we should then see at a glance the relative 
position of these two circles, the points where they intersect one 
another constituting the equinoxes, the points where they are at 
the greatest distance asunder, or the solstices, and various other 
particulars, which, for want of such visible traces, we are now 
obliged to search for by indirect and circuitous methods. It will 
even aid the learner to have Constantly before his mental vision, 
an imaginary delineation of these two important circles on the 
face of the sky. 

155. The method of ascertaining the nature and position of the 
earths orbit, is by observations on the sun's Declination and Right 
Ascension. 

The exact declination of the sun at any time is determined 
from his meridian altitude or zenith distance, the latitude of the 
place of observation being known, (Art. 37.) The instant the 
center of the sun is on the meridian, (which instant is given by 
the transit instrument,) we take the distance of his upper and 
that of his lower limb from the zenith : half the sum of the two 
observations corrected for refraction, gives the zenith distance of 
the center. This result is diminished for parallax, (Art. 84,) and 
we obtain the zenith distance as it would be if seen from the 
center of the earth. The zenith distance being known, the de- 



80 THE SUN. 

.--<*", "- 

clination is readily found, by subtracting that distance from the 
latitude. By thus taking the sun's declination for every day of 
the year at noon, and comparing the results, we learn its motion 

to and from the equator. 
I 

156. To obtain the motion in right ascension, we observe, with 
a transit instrument, the instant when the center of the sun is on 
the meridian. Our sidereal clock gives us the right ascension in 
time (Art. 124,) which we may easily, if we choose, convert into 
degrees and minutes, although it is more common to express right 
ascension by hours, minutes, and seconds. The differences of 
right ascension from day to day throughout the year, give us the 
sun's annual motion parallel to the equator. From the daily re- 
cords of these two motions, at right angles to each other, arran- 
ged in a table,* it is easy to trace out the path of the sun on the 
/artificial globe ; or to calculate it with the greatest precision by 
means of spherical triangles, since the declination and right ascen- 
sion constitute two sides of a right angled spherical triangle, the 
corresponding arc of the ecliptic, that is, the longitude, being the 
third side, (Art. 132.) By inspecting a table of observations, 
we shall find that the declination attains its greatest value on 
the 22d of December, when it is 3 27' 54" south ; that from 
this period it diminishes daily and becomes nothing on the 21st 
of March ; that it then increases towards the north, and reaches 
a similar maximum at the northern tropic about the 22d of June ; 
and, finally, that it returns again to the southern tropic by gra- 
dations similar to those which marked its northward progress. A 
table of observations also would show us, that the daily differences 
of declination are very unequal ; that, for several days, when the 
sun is near either tropic, its declination scarcely -varies at all; 
while near the equator, the variations from day to day are very 
rapid, a fact which is easily understood, when we reflect, that 
at the solstices the equator arid the ecliptic are parallel to each 
other,f both being at , right angles to the meridian; while at the 

* Such a table may be found in Blot's Astronomy, in Delarnbre, and in most collec 
tions of Astronomical Tables. 

t Or, more properly, the tangents of the two circles (which denote the directions of 
the curves at those points) are parallel. 



ANNUAL MOTION. 81 



equinoxes, the ecliptic departs most rapidly from the direction of 
the equator. 

On examining, in like manner, a table of observations of the 
right ascension, we find that the daily differences of right ascen- 
sion are likewise unequal ; that the mean of them all is 3 m 56 s , 
or 236 s , but that they have varied between 215 s and 266 s . On 
examining, moreover, the right ascension at each of the equi- 
noxes, we find that the two records differ by 180; which proves 
that the path of the sun is a great circle, since no other would 
bisect the equinoctial as this does. 

157. The obliquity of the ecliptic is equal to the sun's greatest 
declination. For, by article 22, the inclination of any two great 
circles is equal to their greatest distance asunder, as measured on 
the sphere. The obliquity of the ecliptic may be determined 
from the sun's meridian altitude, or zenith distance, on the day 
of the solstice. The exact instant of the solstice, however, will 
not of course occur when the sun is on the' meridian, but may 
happen at some other meridian ; still, the changes of declination 
near the solstice are so exceedingly small, that but a slight error 
can result from this source. The obliquity may also be found, 
without knowing the latitude, by observing the greatest and least 
meridian altitudes of the sun, and taking half the difference. 
This is the method practiced in ancient times by Hipparchus. 
(Art. 2.) On comparing observations made at different periods 
for more than two thousand years, it is found, that the obliquity 
of the ecliptic is not constant, but that it undergoes a slight dimi- 
nution from age to age, amounting to 52" in a century, or about 
half a second annually. We might apprehend that by successive 
approaches to each other the equator and ecliptic would finally 
coincide ; but astronomers have ascertained by an investigation, 
founded on the principles of universal gravitation, that this varia- 
tion is confined within certain narrow limits, and that the obli- 
quity, after diminishing for some thousands of years, will then 
increase for a similar period, and will thus vibrate for ever about 
a mean value. 

158. The dimensions of the earth's orbit, when compared with its 
own magnitude, are immense. 

11 



2 THE SUN. 

Since the distance of the earth from the sun is 95,000,000 
miles, and the length of the entire orbit nearly 600,000,000 miles, 
it will be found, on calculation, that the earth moves 1,640,000 
miles per day, 68,000 miles per hour, 1,100 miles per minute, and 
nearly 19 miles every second, a velocity nearly fifty times as great 
as the maximum velocity of a cannon ball. A place on the earth's 
equator turns, in the diurnal revolution, at the rate of about 1,000 
miles an hour and fV of a mile per second. The motion around 
the sun, therefore, is nearly 70 times as swift as the greatest mo- 
tion around the axis. 

THE SEASONS. 

159. The change of seasons depends on two causes, (1) the ob- 
liquity of the ecliptic, and (2) the earth's axis always remaining 
parallel to itself. Had the earth's axis been perpendicular to the 
plane of its orbit, the equator would have coincided with the 
ecliptic, and the sun would have constantly appeared in the equa- 
tor. To the inhabitants of the equatorial regions, the sun would 
always have appeared to move in the prime vertical ; and to the 
inhabitants of either pole, he would always have been in the ho- 
rizon. But the axis being turned out of a perpendicular direc- 
tion 23 28', the equator is turned the same distance out of the 
ecliptic ; and since the equator and ecliptic are two great circles 
which cut each other in two opposite points, the sun, while per- 
forming his circuit in the ecliptic, must evidently be once a year 
in each of those points, and must depart from the equator of the 
heavens to a distance on either side equal to the inclination of the 
two circles, that is, 23 28'. (Art. 22.) 

160. The earth being a globe, the sun constantly enlightens 
the half next to him,* while the other half is in darkness. 
boundary between the enlightened and the unenlightened part, is 
called the circle of illumination. When the earth is at one of 
the equinoxes, the sun is at the other, and the circle of illumina- 

* In fact, the sun enlightens a little more than half the earth, since on account of 
his vast magnitude the tangents drawn from opposite sides of the sun to opposite sidea 
of the earth, converge to a point behind the earth, as will be seen by and by in the 
representation of eclipses. The amount of illumination also is increased by refraction. 



THE SEASONS. 



83 



tion passes through both the poles. When the earth reaches one 
of the tropics, the sun being at the other, the circle of illumina- 
tion cuts the earth so as to pass S3 28' beyond the nearer, and 
the same distance short of the remoter pole. These results would 
not be uniform, were not the earth's axis always to remain parallel 
to itself. The following figure will illustrate the foregoing state- 
ments. 

Fig. 31. 




Let ABCD represent the earth's place in different parts of its 
orbit, having the sun in the center. Let A, C, be the position of 
the earth at the equinoxes, and B, D, its positions at the tropics, 
the axis ns being always parallel to itself.* At A and C the sun 
shines on both n and s ; and now let the globe be turned round 
on its axis, and the learner will easily conceive that the sun will 
appear to describe the equator, which being bisected by the hori- 

* The learner will remark that the hemisphere towards n is above, and that towards 
* is below the plane of the paper. It is important to form a just conception of the 
position of the axis with respect to the plane of its orbit. 



84 THE SUN. 

zon of every place, of course the day and night will be equal in all 
parts of the globe.* Again, at B when the earth is at the south- 
ern tropic, the sun shines 23? .beyond the north pole n, and falls 
the same distance short of the south pole s. The case is exactly 
reversed when the earth is at the northern tropic and the sun at 
the southern. While the earth is at one of the tropics, at B for 
example, let us conceive of it as turning on its axis, and we shall 
readily see that all that part of the earth which lies within the 
north polar circle will enjoy continual day, while that within the 
south polar circle will have continual night, and that all other 
places will have their days longer as they are nearer to the en- 
lightened pole, and shorter as they are nearer to the unenlightened 
pole. This figure likewise shows the successive positions of the 
earth at different periods of the year, with respect to the signs, 
and what months correspond to particular signs. Thus the earth 
enters Libra and the sun Aries on the 21st of March, and on the 
21st of June the earth is just entering Capricorn and the sun Can- 
cer. 

161. Had the axis of the earth been perpendicular to the plane 
of the ecliptic, then the sun would always have appeared to move 
in the equator, the days would every where have been equal to the 
nights, and there could have been no change of seasons. On the 
other hand, had the inclination of the ecliptic to the equator been 
much greater than it is, the vicissitudes of the seasons would have 
been proportionally greater than at present. Suppose, for instance, 
the equator had been at right angles to the ecliptic, in which case, 
the poles of the earth would have been situated in the ecliptic 
itself; then in different parts of the earth the appearances would 
have been as follows. To a spectator on the equator, the sun as 
he left the vernal equinox would every day perform his diurnal 
revolution in a smaller and smaller ckcle, until he reached the 
north pole, when he would halt for a moment and then wheel 
about and return to the equator in the reverse order. The pro- 
gress of the sun through the southern signs, to the south pole, 
would be similar to that already described. Such would be the 

* At the pole, the solar disk, at the time of the equinox, appears bisected by the ho- 



FIGURE OP THE EARTH*S ORBIT. 85 

appearances to an inhabitant of the equatorial regions. To a 
spectator living in an oblique sphere, in our own latitude for ex- 
ample, the sun while north of the equator would advance continu- 
ally northward, making his diurnal circuits in parallels further and 
further distant from the equator, until he reached the circle of per- 
petual apparition, after which he would climb by a spiral course 
to the north star, and then as rapidly return to the equator. By a 
similar progre'ss southward, the sun would at length pass the circle 
of perpetual occultation, and for some time (which would be 
longer or shorter according to the latitude of the place of obser- 
vation) there would be continual night. 

The great vicissitudes of heat and cold which would attend 
such a motion of the sun, would be wholly incompatible with the 
existence of either the animal or the vegetable kingdoms, and all 
terrestrial nature would be doomed to perpetual sterility and deso- 
lation. The happy provision which the Creator has made against 
such extreme vicissitudes, by confining the changes of the seasons 
within such narrow bounds, conspires with many other express 
arrangements in the economy of nature to secure the safety and 
comfort of the human race. 



FIGURE OF THE EARTH S ORBIT. . 

162. Thus far we have taken the earth's orbit as a great circle, 
such being the projection of it on the celestial sphere ; but we now 
proceed to investigate its actual figure. 

Were the earth's path a circle, having the sun in the center, the 
sun would always appear to be at the same distance from us ; that 
is, the radius of its orbit, or radius vector, the name given to a line 
drawn from the center of the sun to the orbit of any planet, 
would always be of the same length. But the earth's distance 
from the sun is constantly varying, which shows that its orbit is 
not a circle. We learn the true figure of the orbit, by ascertain- 
ing the relative distances of the earth from the sun at various pe- 
riods of the year. These all being laid down in a diagram, accord- 
ing to their respective lengths, the extremities, on being connected, 
give us our first idea of the shape of the orbit, which appears of 
an oval form, and at least resembles an ellipse ; and, on further 



86 



THE SUN. 



trial, we find that it has the properties of an ellipse. Thus, let E 
(Fig. 32,) be the place of the earth, and a, b, c, &c. successive po- 
sitions of the sun ; the relative lengths of the lines Ea, Eb, &c. be- 
ing known on connecting the points, a, &, c, &c. the resulting 
figure indicates the true shape of the earth's orbit. 




163. These relative distances are found in two different ways ; 
first, by changes in the sun's apparent diameter, and, secondly, by 
variations in his angular velocity. Were the variations in the 
sun's horizontal parallax considerable, as is the case with the 
moon's, this might be made the measure of the relative distances, 
for the parallax varies inversely as the distance, (Art. 82) ; but the 
whole horizontal parallax of the sun is only 9", and its variations 
are too slight and delicate, and too difficult to be found, to serve 
as a criterion of the changes in the sun's distance from the earth. 
But the changes in the sun's apparent diameter, are much more 
sensible, and furnish a better method of measuring the relative 
distances of the earth frorn the sun. By a principle in optics, the 
apparent diameter of an object, at different distances from the 
spectator, is inversely as the distance.* Hence, the apparent 
diameters of the sun, taken at different periods of the year, be- 
come measures of the different lengths of the radius vector. 



* More exactly, the tangent of the apparent diameter is inversely as the distance ; 
but in small angles like those concerned in the present inquiry, the angle itself may bs 
taken for the tangent. 



FIGURE OF THE EARTH'S ORBIT. 87 -^ 

164. The point where the earth, or any planet, in its revolution, 
is nearest the sun, is called its perihelion : the point where it is 
furthest from the sun, its aphelion. The place of the earth's peri- 
helion is known, since there the apparent magnitude of the sun is 
greatest ; and when the sun's magnitude is least, the earth is 
known to be at its aphelion. The sun's apparent diameter when 
greatest is 32' 35."6 ; and when least, 31' 31"; hence the radius 
vector at the aphelion : rad. vector at the perihelion : : 32.5933 : 
31.5167 : : 1.034 : 1. Half of the difference of the two is equal 
to the distance of the focus of the ellipse from the center, a quan- 
tity which is always taken as the measure of the eccentricity of a 
planetary orbit. 

165. The differences of angular velocity in the sun in the dif- 
ferent parts of his apparent revolution, are still more remarkable. 
At the perihelion, the sun moves in twenty-four hours over an arc 
of 61', while at the aphelion he describes in the same time an arc 
of only 57', these being the daily increments of longitude in those 
two points respectively. If the apparent motions of the sun de- 
pended alone on our different distances from him, the angular ve- 
locity would vary inversely as the distance, and the ratio expressed 
by these two numbers would be the same as that of the two num- 
bers which denote the differences of apparent diameter in these 



two points. That is, (=1.07) would equal (=1.034) ; 
57 31.5167 

but the first fraction is equal to the square of the second, for 1.07= 
1 .034 2 . Hence, the sun's angular velocities are to each other inversely 
as the squares of the distances at the perihelion and the aphelion ; and 
by a similar method, the same is found to be true in all points of 
the revolution. 

The angular velocities, therefore, which can be measured very 
accurately by the daily differences of right ascension and declina- 
tion (Art. 132,) converted into corresponding longitudes, enable 
us to determine the different distances of the earth from the sun 
at various points in the orbit. 

166. Since the arcs described by the earth in any small times, 
as in single days, are inversely as the squares of the distances, con- 



88 



THE SUN. 



r 



sequently, the distances are inversely as the square roots of the arcs. 
Upon this principle, the relative distances of the earth from the 
sun, in every point of its revolution, may be easily calculated. 
Thus, we have seen that the arcs described by the sun in one day 
at the perihelion and aphelion are as 61 to 57. Hence the distances 
of the earth from the sun at those two points are as \/57 to \/6i, 
or as 1 to 1.034. From twenty-four observations made with the 
greatest care by Dr. Maskelyne at the Royal Observatory of 
Greenwich, the following distances of the earth from the sun are 
determined for each month in the year. 



Time of Observation. Distances. 

January 12-13, 0.98448 

February 17-18, 0.98950 

March 14-15, 0.99622 

April 28-29, 1.00800 

May 15-16, 1.01234 

June 17-18, 1.01654 



Time of Observation. Distances. 

July 18-19, 1.01658 

August 26-27, 1.01042 

September 22-23, 1.00283 

October 24-25, 0.99303 

November 18-20, 0.98746 

December 17-18, 0.98415 



Fig. 33. 



B 



167. The angular velocity being 
inversely as the square of the distance 
in all parts of the solar orbit, it follows 
that the product of the angle described 
in any given time, by the square of the 
distance, is always the same constant 
quantity. For if of two factors, A x 
B, A is increased as B is diminished, 
the product of A and B is always the 
same. If, therefore, from the sun S 
(Fig. 33,) two radii be drawn to T, 

B, the extremities of the arc described in one day, then ST 2 xTB 
gives the same product in all parts of the orbit.* 

/ 

168. The radius vector of the solar orbit describes equal spaces 
in equal times, and in unequal times, spaces proportional to the times. 

Let TB (Fig. 33,) be the arc described by the sun in one day ; 
then, Sector TSB=iSB xTB. 




* TB, as seen from the earth, would be projected into a circular arc, equal to the 
measure of the angle at S. 



FIGURE OF THE EARTH'S ORBIT. 89 

Taking Sb as any radius, describe the circular arc ab, which is 
.he measure of the angle at S. Now, 

Sb : ab : : SB : BT=SBx^- ; and substituting this value of BT 



in the above equation, we have TSB=!SBxSBx =iSB 2 x. 

&o feo 

But Sb is constant, and the product of SB 2 x& is likewise constant ; 
therefore the sector is always equal to a constant quantity, and 
therefore the radius vector passes over equal spaces in equal 
times.* 

The sun's orbit may be accurately represented by taking some 
point as the perihelion, drawing the radius vector to that point, 
and, considering this line as unity, drawing other radii making 
angles with each other such that the included areas shall be pro- 
portional to the times, and of a length required by the distance of 
each point as given in the table (Art. 166.) On connecting these 
radii, we shall thus se'e at once how little the earth's orbit departs 
from a perfect circle. Small as the difference appears between 
the greatest and least distances, yet it amounts to nearly ? of the 
perihelion distance, a quantity no less than 3,000,000 of miles. 

169. The foregoing method of determining the figure of the 
earth's orbit is founded on observation ; but this figure is subject 
to numerous irregularities, the nature of which cannot be clearly 
understood without a knowledge of the leading principles of Uni- 
versal Gravitation. An acquaintance with these will also be in- 
dispensable to our understanding the causes of the numerous ir- 
regularities, which (as will hereafter appear) attend the motions 
of the moon and planets. To the laws of universal gravitation, 
therefore, let us next apply our attention. 



* Francoeur, Uran., p. 62. 

12 



CHAPTER III. 

OF UNIVERSAL GRAVITATION. 

170. UNIVERSAL GRAVITATION, is that influence by which every 
body in the universe, whether great or small, tends towards every 
other, with a force which is directly as the quantity of matter, and 
inversely as the square of the distance. 

As this force acts as though bodies were drawn towards each 
other by a mutual attraction, the force is denominated attraction ; 
but it must be borne in mind, that this term is figurative, and im- 
plies nothing respecting the nature of the force. 

The existence of such a force in nature was distinctly asserted 
by several astronomers previous to the time of Sir Isaac Newton, 
but its laws were first promulgated by this wonderful man in his 
Principia, in the year 1687. It is related, that while sitting in a 
garden, and musing on the cause of the falling of an apple, he 
reasoned thus ;* that, since bodies far removed from the earth fall 
towards it, as from the tops of towers, and the highest mountains, 
why may not the same influence extend even to the moon ; and 
if so, may not this be the reason why the moon is made to revolve 
around the earth, as would be the case with a cannon ball were 
it projected horizontally near the earth with a certain velocity. 
According to the first law of motion, the moon, if not continually 
drawn or impelled towards the earth by some force, would not 
revolve around it, but would proceed on in a straight line. But 
going around the earth as she does, in an orbit that is nearly cir- 
cular, she must be urged towards the earth by some force, which, 
in a given time, may be represented by the versed sine of the arc 
described in that time. For let .the earth (Fig. 34,) be at E, and 
let the arc described by the moon in one second of time be Ab. 
Were the moon influenced by no extraneous force, to turn her 
aside, she would have described, not the arc Ab, but the straight 
line AB, and would have been found at the end of the given time 

* Pemberton's View of Newton's Philosophy. 



UNIVERSAL GRAVITATION. 



91 




at B instead of b. She therefore departs from J;he line in which 
she tends naturally to move, by the line B&, which in small angles 
may be taken as equal to the versed sine Aa. This deviation 
from the tangent must be owing to 
some extraneous force. Does this force t 
correspond to what the force of gravi- 
ty exerted by the earth, would be at the 
distance of the moon? Now we know the 
distance of the moon from the earth, and 
of course the circumference of her orbit. 
We also know the time of her revolu- 
tion around the earth. Hence we may 
estimate the length of the arc Kb de- 
scribed in one second ; and knowing 
the arc, we can calculate its versed sine. 
For the moon being 60 times as far from the center of the earth, 
as the surface of the earth is from the center, consequently, since 
the force of gravity decreases as the square of the distance in- 
creases,* the space through which the moon would fall by the 

force of the earth's attraction alone, would be p = .05 inches. 

On calculating the value of the versed sine of the arc described in 
one second, it proves to be the same. Hence gravity, and no other 
force than gravity, causes the moon to circulate around the earth. 

171. By this process it was discovered that the law of gravita- 
tion extends to the moon. By subsequent inquiries it was found 
to extend in like manner to all the planets, and to every member 
of the solar system ; and, finally, recent investigations have shown 
that it extends to the fixed stars. The law of gravitation, there- 
fore, is now established as the grand principle which governs all 
the motions of the heavenly bodies. Hence, nothing can be more 
deserving of the attention of the student, than the development of 
the results of this universal law. A few of them only are all that 
can be exhibited in a work like the present : their full develop- 



* Natural Philosophy, Art. 7. That gravity follows the ratio of the inverse square 
of the distance was, however, inferred by Newton from one of Kepler's Laws, to be 
mentioned hereafter. 



92 



UNIVERSAL, GRAVITATION. 



ment must be sought for in suqfc great works as the Mecanique 
Celeste of La Place. 

172. If a ~body revolves about an immovable center of force* 
and is constantly attracted to it, the radius vector will always 
move in the same plane, and describe areas about the center 
proportional to the times* 

Let S (Fig. 35,) be the center of force, and suppose a body to 
be projected at P in the direction of PQR, and take PQ=QR; 
then, by the first law of motion, the body would move uniformly 
in the direction PQR, and describe PQ, QR, in the same time, if 
no other force acted upon it. But when the body comes to Q, 

Fig. 35. 



Tsi 




let a single impulse act at S, sufficient to draw the body through 
QV, in the time it would have described QR ; and complete the 
parallelogram VQRC, and the body in the same time will describe 
QC ; therefore, PQ, QC, are described in the same time. But 
the triangle SCQ=SRQ=SPQ ; that is, equal areas are described 
in equal times. For the same reason, if a single impulse act at 
C, D, E, &c. at equal intervals of time, the several areas SPQ, 
SQC, SCD, SDE, &c. will all be equal to each other. Now this 

* The learner will remark that what has been before proved (Art. 168,) respecting 
the radius vector of the earth, is here shown to hold good with respect to every body 
which revolves around a center of force ; and the same is true of several other propo- 
sitions demonstrated in this chapter. 



UNIVERSAL GRAVITATION. 



93 



demonstration is independent of any particular dimensions in the 
several triangles, and consequently holds good when they are 
taken indefinitely small, in which case we may consider the force 
as acting, not by separate impulses, Hut constantly, causing the 
body to describe a curve around S. And as no force acts out of 
the plane SPQ, the whole curve must lie in that plane ; that is, 
the body moves always in the same plane. 

173* If a body describes a curve around a center towards which it 
tends by any force, the angular velocity of the body around that center 
is reciprocally as the square of the distance from it.* 

Let ABE (Fig. 36,) be any curve de- Fig. 36. , 

scribed about the center S ; draw SA, SB, 
to any two points of the curve A and B ; 
and let AD, BE, be described in indefi- 
nitely small equal times. Join SD and 
SE, and with the center S and distance 
SD, describe a circle meeting SA, SB, SE, 
in F, G, H ; and with the center S and 
distance SE describe a circle meeting SB 

T7" 

in K. 

Because AD and BE are described in 
equal times, the trianglos ASD, BSE, are 
equal. Hence, 

DF : EK :: BS : ASf :: BS 2 : BSxAS (1) 
GH : EK :: SH : SE :: SF : SE :: SA : SB 

Hence, (1) DF : BS 2 :: EK : BSxAS 
(2)GH:AS 2 ::EK:BSxAS 
.-. DF:GH::BS 2 : AS 2 . 

But DF and GH measure the respective angular velocities at 
A and B, while AS and BS represent the distance at the same 
points. Therefore the angular velocities are reciprocally as the 
squares of the distances. J 

174. In the same curve, the velocity, at any point of the curve, 

* It will be remarked that this is a general proposition, of which article 165 affords 
a particular example. 

t DF and EK are considered as the altitudes of the triangles respectively. 
T Stewart's Phys. and Math. Essays. 




SA 2 : BSxAS (2) 



94 UNIVERSAL GRAVITATION. 

varies inversely as the perpendicular drawn from the center of 
force to the tangent at that point. 

Draw SY (Fig. 35,) perpendicular to QP produced ; then the 
area SPQ=|PQ x SY, which varies as PQ x SY /. PQ x 

Now 



in the curve described from P, with a constant force, SY becomes 
a perpendicular to the tangent to the curve. But by article 
] 72, the area described in a given time is constant. Therefore 

SPQ is constant, and V cr - ; that is, the velocity varies inverse- 

SY 

ly as the perpendicular upon the tangent. Hence, the velocity of 
a revolving body increases as it approaches the center of force. 

175. If equal areas be described about a center in equal times* 
the force must tend towards that center. 

Let SPQ (Fig. 35,)=SQC ; now SPQ-SQR .-. SQC-SQR/. 
CR is parallel to QS. Complete the parallelogram QRCV, and 
by the supposition the body describes QC, in consequence of the 
impulse at Q, and it would have described QR if no such impulse 
had acted ; therefore QV must represent that motion impressed 
at Q, which, in conjunction with the motion QR, can make a body 
describe QC, and QV is directed to S. 

176. Now it appears from article 168, that it is a fact, derived 
from observation, that the earth's radius vector describes equal 
areas in equal times ; and by similar observations the same is 
found to be true of each of the primary planets about the sun, 
and of each of the satellites about its primary. Hence, it is in- 
ferred, that the primary planets all gravitate towards the sun, and 
that the secondary planets all gravitate towards their respective 
primaries. 

It has further been established by observation, (Art. 162,) that 
the planetary orbits are ellipses ; and hence the application of the 
principles of gravitation, so far as respects the sun and planets, 
may be confined to the consideration of the motion of a body in 
an elliptical orbit. 

177. The distance of any planet from the sun at any point in its 



UNIVERSAL GRAVITATION. 



95 



orbit, is to its distance from the superior focus, as the square of its 
velocity at its mean distance from the sun, is to the square of its ve- 
locity at the given point. 

Let ADBE (Fig. 37,) be the orbit of a planet, S the focus in 
which the sun is placed, AB the transverse and DE the conjugate 
axis, C the center, and F the superior focus. Let the planet be 
any where at P ; and dr aw a tangent to the orbit at P, on which 
from the foci let fall the perpendiculars SG, FH. Draw also DK 
touching the orbit in D, and let SK be perpendicular to it. Let 

Fig. 37. 

N 




the velocity of the planet when at the mean distance at D=C, and 
when at P=V. Join SP, FP. Then (Art. 174,) the velocity at 
D is to the velocity at P, as SG to SK ; that is, 

C:V::SG:DC. 
C 2 : V 2 : : SG 2 : DC 2 . 

But because the triangles SGP, FHP, are equiangular, having 
right angles at G and H, and also, from the nature of the ellipse, 
the angles SPG, FPH, equal, 

SP : PF : : SG : FH : : SG 2 : CD 2 =FHxSG 
.-. SP:PF::C 2 :V 2 

178. If of two bodies gravitating to the same center, one descends 
in a straight line, and the other revolves in a curve ; then, if the ve- 
locities of these bodies are equal in any one case, when they are 



96 



UNIVERSAL GRAVITATON. 



equally distant from the center, they will always be equal when they 
are equally distant from it. 

Let ABC (Fig. 38,) be a curve which a body Fig> 38 ' 

describes about a center Sto which it gravi- 
tates, while another body descends in a 
straight line AS to that center. Let BC be 
any arc of the curve ABC, and let BD, CH, 
be arcs of circles described from the center 
S, intersecting the line AS in D and H. 
From the center S describe the arc bd, in- 
definitely near to BD, and draw E/ perpen- 
dicular to B6. Then, because the distances 
SD and SB are equal, the forces of gravity 
at D and B are also equal. Let these forces 
be expressed by the equal lines ~Dd and BE ; 
and let the force BE be resolved into the 
forces E/* and B/*. The force E/, acting at 
right angles to the path of the body, will not affect its velocity in 
that path, but will only draw it aside from a rectilinear course and 
make it proceed in the curve B6C. But the other force B/*, acting 
in the direction of the course of the body, will be wholly employed 
in accelerating it. And because B and b are indefinitely near to 
each other, and likewise D and d, the accelerating force from B to 
b and from D to d, may be considered as acting uniformly. 
Therefore, the accelerations of the bodies in D and B, produced 
in equal times, are as the lines Dd, B/*; and hence, putting d for 
the increment of velocity at d, and /for the increment of velocity 
at/, 

d:/::DdorBE:B/. (1) 

And because the angle at E is a right angle, 




Hence, BE : B/: : v/B6 : v/B/ (2) 
And, (1) and (2),d:f: : x/B&WB/ (3) 

But, putting b for the velocity at b, and observing that, in falling 
bodies, the velocities are as the square roots of the spaces, 



Therefore, (3) and (4), b:f::d:f.-. b=d ; that is, the velocity at 
b equals the velocity at d. And, since the same reasoning holds 



UNIVERSAL GRAVITATION. 97 

for successive points that may be taken at equal distances from B 
and D, therefore, if of two bodies, &c.* 

179. The law according to which the planets gravitate is such, that 
any body under the influence of the same force, and falling direct to 
the sun, will have its velocity at any point equal to a constant velocity 
multiplied into the square root of the distance it has fallen through, 
divided by the square root of the distance between the body and the 
9im's center. 

Suppose a planet to revolve in the elliptical orbit APB (Fig. 37); 

at A, the higher apsis, the velocity V=C (^?)* (Art. 177) ; or 



if AN, in the axis produced=AF, v=c - Let a bo <ty at 

A begin to descend towards S with this velocity, then if SL=SP,- 
the velocity of the planet at P will be the same as that of the fall- 
ing body at L, (Art. 178.) But the velocity of the planet at P is 

this velocity is equal to the constant ve- 



( p~ r 



locity expressed by C, multiplied into the square root of NL, the 
distance fallen through,J divided by the square root of LS, the 
distance between the body and the sun's center. 

180. The force with which any planet gravitates to the sun, is in- 
versely as the square of its distance from the sun's center. 

Let C (Fig. 39,) be the center to which the falling body gravi- 
tates, A the point from which it begins to fall, and its velocity at 
any point B, is to its velocity in the point G, which bisects AC, as 



Let I)EF be a curve such that if AI) be an ordinate 

or a perpendicular to AC, meeting the curve in D, and BE any other 

* Principia, Lib. i, Pr. 40. Stewart's Math, and Phys. Essays, Pr. 13. 
t For SN=AB=SP-f-PF=SP-[-NL '. PF=NL. 

* That NL ( PF) is the distance fallen through to acquire the velocity at P, is de- 
monstrated by writers on Central Forces. (See Vince, Syst. Ast., Art. 823.) 

Playfair, Phys. Ast. 

jj For, denoting the velocity at B by V, and the velocity at G by V ; t 
ABU t /AGU . m /AB\ * . /AGU . . /ABU . 

J : IGC) "(BC) ' L 
13 



98 



UNIVERSAL GRAVITATION. 



ordinate, AD is to BE as the force at A to the force at B, then 



will twice the area ABED be equal to the 
square of the velocity which the body has 
acquired in B.* If therefore the velocity 
at B be V, that at the middle point G being c, 



Fig. 39. 



A 



V=c , and therefore SABED-c 2 . ; 

VijU/ >(_/ 

and since AB = AC - BC, 2ABED = c 2 . 

AC-BC 2 /AC \ . 

^ =c 2 1 ._ ^ 1 1. For the same reason, 
JoO VJoU / 

if be be drawn indefinitely near to BE, 2A&eD 

(AC \ 
-r-^ 1 1, and therefore the difference of 
bC 1 

these areas, or 2BbeE, that is, SEBxB&^c 2 
2 AC(BC-6C) 2 ACxB& 





G 



c 



=C 2.^. orEB=c 2 



AG 

BC S; 



now c 2 and AG are constant 



quantities, therefore EB varies inversely as BC 2 . But EB repre- 
sents the force of gravity at B, and BC the distance from the 
sun. Therefore, the force of gravity of a planet in different parts 

of its orbit, is inversely as the square of its distance from the sun. 



181. The line CG is the same with the mean distance of the 
planet in an orbit of which AC is the length of the transverse axis ; 
and if the gravitation at that distance =F, and the mean distance 

itself=a, then since EB=c 2 ^-, F=c 2 x~=-, or aF=c 2 . 



BC S 



a* a 



* This principle is demonstrated by the aid of Fluxions as follows : I;' ' 

By construction, BE is proportional to the force at B=:^-, v being the velocity 

which the moving body has acquired at B, and t the time of the descent from A to B. 
Now B6 is the momentary increment of B A the space, and therefore vdt ; therefore 
BExB6:=?5afo. And 2BExBJ=2ucfa>. But BE XB6 is the momentary increment of 
the area ABED, and 2rdo is the momentary increment of r 2 ; therefore the square of 
the velocity of the moving body, and twice the area of ABED, increase at the same 
rate, and begin to exist at the same time ; therefore they are equal. (See Playfair's 
Outlines, Mechanics, Art. 96.) 
t fiC being ultimately equal to BC. 



UNIVERSAL GRAVITATION. 99 

182. The squares of the times of revolution of any two planets* 
are as the cubes of their mean distances from the sun. 

If a be the mean distance, or the semi-transverse axis, b the 
semi-conjugate, then tf#&=area of the orbit.* But as c is the ve- 
locity at the mean distance, or the elliptic arch which the planet 
moves over in a second when it is at D, (Fig. 37.) the vertex of the 
conjugate axis, therefore %bc is the area described in that second 
by the radius vector ; and since the area is the same for every 
second of the planet's revolution (Art. 172,) therefore the area of 
the orbit divided by ?bc will give the number of seconds in 

which the revolution is completed, which= j-r- = - ; or, since 

%bc c 



c 2 = F, (Art. 181,) the time of a revolution = ==. 

VaF 

Hence, let t, t', be the times of revolutions for two different plan- 
ets, of which the mean distances are #, a', and the force of gravity 

at those distances F, P. Then t: t' : : 2* \/4- : 2*\/V : 



But (Art. 1800 



--, or 2 : t : : a 3 : ct' 3 . That is, the squares of the times are as the 

cubes of the mean distances ; or, since the major axes of the or- 
bits are double the mean distances, the squares of the times are as 
the cubes of the major axes. 

183. This is one of Kepler's three great Laws, which, taken in 
connexion, are as follows : 

1. The orbits of all the planets are ellipses, the sun occupying the 
common focus. (Art. 176.) 

2. The radius vector of any planet describes areas proportional 
to the times. (Art. 172.) 

3. The squares of the periodical times are as the cubes of the ma- 
jor axes of the orbits. (Art. 182.) 

These great and fundamental principles of the planetary mo- 
tions, were discovered by the illustrious Kepler by long and as- 
siduous study of the observations made by Tycho Brahe, and 

* Day's Mensuration. 



100 UNIVERSAL GRAVITATION. 

hence he has been called the legislator of the skies. They, there 
fore, became known as facts, before they were demonstrated 
mathematically. The glory of this achievement was reserved 
for Newton, who proved that they were necessary results of the 
law of universal gravitation. 

MOTION IN AN ELLIPTICAL ORBIT. 

184. Having now acquired some knowledge of the law of uni- 
versal gravitation, let us next endeavor to gain a just conception 
of the forces by which the planets are made to revolve in their 
orbits about the sun. In obedience to the first law of motion, 
every moving body tends to move in a straight line ; and were not 
the planets deflected continually towards the sun by the force of 
attraction, these bodies as well as others would move forward in 
a rectilineal direction. We call the force by which they tend to 
such a direction the projectile force, because its effects are the 
same as though the body were originally projected from a certain 
point in a certain direction. It is an interesting problem for me- 
chanics to solve, what was the nature of the impulse originally 
given to the earth, in order to impress upon it its two motions, the 
one around its own axis, the other around the sun ? If struck in 
the direction of its center of gravity it might receive a forward 
motion, but no rotation on its axis. It must, therefore, have been 
impelled by a force, whose direction did not pass through its cen- 
ter of gravity. Bernouilli, a celebrated mathematician, has calcu- 
lated that the impulse must have been given very nearly in the 
direction of the center, the point of projection being only the 165th 
part of the earth's radius from the center.* This impulse alone 
would cause the earth to move in a right line : gravitation towards 
the sun causes it to describe an orbit. Thus a top spinning on a 
smooth plane, as that of glass or ice, if impelled in a direc- 
tion not passing through the center of gravity, may be made to 
imitate the two motions of the earth, especially if the experiment 
is tried in a concave surface like that of a large bowl. The re- 
sistance occasioned by the surface on which the top moves, and 

* Francceur, Uran. p. 49. 



UNIVERSAL GRAVITATION. 



101 



that of the air, will generally destroy the force of projection and 
cause the top to revolve in a smaller and smaller orbit ; but the 
earth meets with no such resistance, and therefore makes both her 
days and years of the same length from age to age. A body, 
therefore, revolving in an orbit about a center of attraction, is 
constantly under the influence of two forces, the projectile force, 
which tends to carry it forward in a straight line which is a tan- 
gent to its orbit, and the centripetal force, by which it tends to- 
wards the center. 

185. The most simple example we have of the combined action 
of these two forces is the motion of a missile ;ti\rJoJvCn/ from; thqi 
hand, or of a ball fired from a cannon. It js welf known 'that fhe 
particular form of the curve described by thej^cjjectfle,\ii> krttteT 1 
case, will depend upon the velocity with which it is thrown. In 
each case the body will begin to move in the line of direction in 
which it is projected, but it will soon be deflected from that line 
towards the earth. It will however continue nearer to the line of 
projection as the -velocity of projection is greater. Thus let AB 

Fig. 40. 




(Fig. 40,) perpendicular to AC represent the line of projection. 
The body will, in every case, commence its motion in the line AB 
which will therefore be the tangent to the curve it describes ; but 
if it be thrown with a small velocity, it will soon .depart from the 
tangent, describing the line AD ; with a greater velocity it will 
describe a curve nearer to the tapgent, as AE ; and with a still 
greater velocity it will describe the curve AF. 

As an example of a body revolving in an orbit under the influ- 
ence of two forces, suppose a body placed at any point P (Fig. 40') 
above the surface of the earth, and let PA be the direction of the 
earth's center ; that is, a line perpendicular to the horizon. If the 



102 UNIVERSAL GRAVITATION. 

Fig. 40'. 




body were allowed to move without receiving any impulse, it 
would d'escend to &e*e&rth in the direction PA with an accelerated 
motion. '' But 'suppose that, at the moment of its departure from 
P, it receives a blow in the direction PB, which would carry it to 
B in the time the body would fall from P to A ; then, under the in- 
fluence of both forces, it would descend along the curve PD. If 
a stronger blow were given to it in the direction PB, it would de- 
scribe a larger curve, PE ; or, finally, if the impulse were suffi- 
ciently strong, it would circulate quite around the earth, and re- 
turn again to P, describing the circle PFG. With a velocity of 
projection still greater, it would describe an ellipse, PIK ; and if 
the velocity were increased to a certain degree, the figure would 
become a parabola or hyperbola LMP, and never return into 
itself. 

186. In figure 41, suppose the planet to have passed the point C 
with so small a velocity, that the attraction of the sun bends its 
path very much, and causes it immediately to begin to approach 
towards the sun ; the sun's attraction will increase its velocity as 
it moves through D, E, and F. For the sun's attractive force on 
the planet, when at D, is acting in the direction DS, and, on account 
of the small inclination of DE to DS, the force acting in the line 
DS helps the planet forward in the path DE, and thus increases 
its velocity. In like manner the velocity of the planet will be con- 
tinually increasing as it passes through D, E, and F ; and though 
the attractive force, on account of the planet's nearness, is so much 
increased, and tends, therefore, to make the orbit more curved, 



UNIVERSAL GRAVITATION. 



103 




yet the velocity is also so much increased, that the orbit is not 
more curved than before. The same increase of velocity occa- 
sioned by the planet's approach to the sun, produces a greater in- 
crease of centrifugal force which carries it off again. We may 
see also why, when the planet has Fig. 41. 

reached the most distant parts of its 
orbit, it does not entirely fly off, and 
never return to the sun. For when 
the planet passes along H, K, A, the 
sun's attraction retards the planet, 
just as gravity retards a ball rolled up 
hill ; and when it has reached C, its 
velocity is very small, and the attrac- 
tion at the center of force causes a 
a great deflection from the tangent, 
sufficient to give its orbit a great cur- 
vature, and the planet turns about, returns to the sun, and goes 
over the same orbit again.* As the planet recedes from the sun, 
its centrifugal force diminishes faster than the force of gravity, so 
that the latter finally preponderates, f 

187. We may imitate the two motions of the earth, the diurnal 
and the annual, in the following manner. Suspend from the ceiling 
of a room, by a string long enough to reach to the level of the 
eye, a ball (of wood for example) four or five inches in diameter, to 
represent the earth. In the point occupied by the ball when at 
rest, let a small globe be supported to represent the sun. The sus- 
pended ball being drawn out of its place of rest, which is directly 
under the point of suspension, it will tend constantly towards the 
same point, by a force which corresponds to the force of attraction 
of a central body. If, when thus drawn out, it be impelled by a 
blovx in the direction of the center of gravity, it will revolve with- 
out turning on its axis ; but if struck out of the center of gravity, 
it will, at the same time, revolve on its axis and in its orbit. 



Airy. 

t The centrifugal force varies inversely as the cube of the distance, while the force 
of gravity is inversely as the square. The centrifugal force, therefore, increases faster 
than the force of gravity as a body is approaching the sun, and decreases faster as the 
body recedes from the sun. (See M. Stewart's Phys. and Math. Tracts, Prop. 8.1 



CHAPTER IV. 

PRECESSION OF TI$B EQUINOXES NUTATION ABERRATION MOTION 

OF THE APSIDES MEAN AND TRUE PLACES OF THE SUN. 

188. THE PRECESSION OF THE EQUINOXES, is a slow but continual 
shifting of the equinoctial points from east to west. 

Suppose that we mark the exact place in the heavens, where, 
during the present year, the sun crosses the equator, and that this 
point is close to a certain star ; next year the sun will cross the 
equator a little way westward of that star, and so every year a 
little further westward, until, in a long course of ages, the place 
of the equinox will occupy successively every part of the ecliptic, 
until we come round to the same star again. As, therefore, the 
sun, revolving from west to east in his apparent orbit, comes 
round towards the point where it left the equinox, it meets the 
equinox before it reaches that point. The appearance is as though 
the equinox goes forward to meet the sun, and hence the phenom- 
enon is called the Precession of the Equinoxes, and the fact is 
expressed by saying that the equinoxes retrograde on the ecliptic, 
until the line of the equinoxes makes a complete revolution from 
east to west. The equator is conceived as sliding westward on 
the ecliptic, always preserving the same inclination to it, as a ring 
placed at a small angle with another of nearly the same size, 
which remains fixed, may be slid quite around it, giving a cor- 
responding motion to the two points of intersection. It must be 
observed, however, that this mode of conceiving of the precession 
of the equinoxes is purely imaginary, and is employed merely for 
the convenience of representation. 

189. The amount of precession annually is 50." 1 ; whence, 
since there are 3600" in a degree, and 360 in the whole circum- 
ference, and consequently, 1296000", this sum divided by 50.1 
gives 25868 years for the period of a complete revolution of the 
equinoxes. 



PRECESSION OF THE EQUINOXES. 105 

190 Suppose now we fix to the center of each of the two 
rings (Art. 188) a wire representing its axis, one corresponding to 
the axis of the ecliptic, the other to that of the equator, the ex- 
tremity of each being the pole of its circle. As the ring deno- 
ting the equator turns round on the ecliptic, which with its axis 
remains fixed, it is easy to conceive that the axis of the equator 
revolves around that of the ecliptic, and the pole of the equator 
areund the pole of the ecliptic, and constantly at a distance equal 
to the inclination of the two circles. To transfer our conceptions 
to the celestial sphere, we may easily see that the axis of the diur- 
nal sphere, (that of the earth produced, Art. 28,) would not have 
its pole constantly in the same place among the stars, but that this 
pole would perform a slow revolution around the pole of the 
ecliptic from east to west, completing the circuit in about 26,000 
years. Hence the star which we now call the pole star, has not 
always enjoyed that distinction, nor will it always enjoy it here- 
after. When the earliest catalogues of the stars were made, this 
star was 12 from the pole. It is now 1 24', and will approach 
still nearer ; or, to speak more accurately, the pole will come still 
nearer to this star, after which it will leave it, and successively 
pass by others. In about 13,000 years, the bright star Lyra, 
which lies on the circle of revolution opposite to the present pole 
star, will be within 5 of the pole, and will constitute the Pole 
Star. As Lyra now passes near our zenith, the learner might 
suppose that the change of position of the pole among the stars, 
would be attended with a change of altitude of the north pole 
above the horizon. This mistaken idea is one of the many mis- 
apprehensions which result from the habit of considering the 
horizon as a fixed circle. in space. However the pole might shift 
its position in space, we should still be at the same distance from 
it, and our horizon would always reach the same distance be- 
yond it. 

191. The precession of the equinoxes is an effect of the spheroidal 
figure of the earth, and arises from the attraction of the sun and 
moon upon the excess of matter about the earth's equator. 

Were the earth a perfect sphere the attractions of the sun and 
moon upon the earth would be in equilibrium among themselves. 

14 



106 



THE SUN. 



But if a globe were cut out of the earth, (taking half the polar 
diameter for radius,) it would leave a protuberant mass of matter 
in the equatorial regions, which may be considered as all collected 
into a ring resting on the earth. The sun being in the ecliptic, 
while the plane of this ring is inclined to the ecliptic 23 28', of 
course the action of the sun is oblique to the ring, and may be 
resolved into two forces, one in the plane of the equator, and the 
other perpendicular to it. The latter only can act as a disturbing 
force, and tending as it does to draw down the ring to the ecliptic, 
the ring would turn upon the line of the equinoxes as upon a 
hinge, and dragging the earth along with it, the equator would 
ultimately coincide with the ecliptic were it not for the revolution 
of the earth upon its axis. This may be better understood by the 
aid of a diagram. Let TAB (Fig. 42,) represent the equator, 

Fig. 42. 




TED the ecliptic, and AD the solstitial colure. Let AB be the 
movement of rotation for a very short time, being of course in the 
order of the signs and in the direction of the equator. Let BC be 
the movement produced by the disturbing force of the sun in the 
same time. The point A will describe the diagonal AC, the equa- 
tor will take the inclined situation CAT' ; the equinoctial point 
will' retrograde from T to T' ; the colure AD will take the posi- 
tion AE, while the inclination of the two planes, that is, the ob- 
liquity of the ecliptic, will remain nearly the same.* 

192. The moon conspires with the sun in producing the pre- 
cession of the equinoxes, its effect, on account of its nearness to 
the earth, being more than double that of the sun, or as 7 to 3. 
The planets likewise, by their attraction, produce a small effect 

Delambre, t. 3, p. 145. Playfair's Outlines, 2, 308. 



PRECESSION OF THE EQUINOXES. 10? 

upon the equatorial ring, but the result is slightly to diminish the 
amount of precession. The whole effect of the sun and moon 
being 50. "41, that of the planets is 0.31, leaving the actual amount 
of precession 50."!.* 

This effect is not to be imagined as taking place merely at the 
time of the solstices, but as resulting constantly from the action 
of the sun and moon on the equatorial ring, and at every revolu- 
tion of this ring along with the earth on its axis. Conceive of 
any point in the ring, and follow it round in the diurnal revolution, 
and it will be seen that that point, in consequence of the attrac- 
tion of the sun and moon, will be made to cross the ecliptic a little 
further westward than on the preceding day. 

193. The time occupied by the sun in passing from the equinoc- 
tial point round to the same point again, is called the TROPICAL YEAR. 
As the sun does not perform a complete revolution in this inter- 
val, but falls short of it 50.**1, the tropical year is shorter than the 
sidereal by 20m. 20s. in mean solar time, this being the time of 
describing an arc of 50." 1 in the annual re volution, f The 
changes produced by the precession of the equinoxes in the ap- 
parent places of the circumpolar stars, have led to some interest- 
ing results in chronology. In consequence of the retrograde mo- 
tion of the equinoctial points, the signs of the ecliptic (Art. 35,) 
do not correspond at present to the constellations which bear the 
same names, but lie about one whole sign or 30 westward of 
them. Thus, that division of the ecliptic which is called the sign 
Taurus, lies in the constellation Aries, and the sign Gemini in the 
constellation Taurus. Undoubtedly, however, when "the ecliptic 
was thus first divided, and the divisions named, the several con- 
stejlations lay in the respective divisions which bear their names. 
How long is it, then, since our zodiac was formed ? 

50."1 : 1 year :: 30(=108000") : 2155.6 years. 

The result indicates that the present divisions of the zodiac 
were made soon after the establishment of the Alexandrian school 
of astronomy. (Art 2.) 

* Francoeur, Uran. 162. t 59' 8."3 : 24h. : : 50.1 : 20m. 20s. 



108 THE SUN. 



NUTATION. 

194. NUTATION is a vibratory motion of the earth 1 s axis, arising 
from periodical ^fluctuations in the obliquity of the ecliptic. 

If the sun and moon moved in the plane of the equator, there 
would be no precession, and the effect of their action in producing 
it varies with their distance from that plane. Twice a year, there- 
fore, namely, at the equinoxes, the effect of the sun is nothing ; 
while at the solstices the effect of the sun is a maximum. On 
this account, the obliquity of the ecliptic is subject to a semi-an- 
nual variation, since the sun's force which tends to produce a 
change in the obliquity is variable, while the diurnal motion of 
the earth which prevents the change from taking place, is con- 
stant. Hence the plane of the equator is subject to an irregular 
motion which is called the Solar Nutation. The name is derived 
from the oscillatory motion communicated by it to the earth's axis, 
while the pole of the equator is performing its revolution around 
the pole of the ecliptic (Art. 190.) The effect of the sun however 
is less than that of the moon, in the ratio of 2 to 5. By the nuta- 
tion alone the pole of the earth would perform a revolution in a 
very small ellipse, only 18" in diameter, the center being in the 
circle which the pole describes around the pole of the ecliptic ; 
but the combined effects of precession and nutation convert the 
circumference of this circle into a wavy line. The motion of the 
equator occasioned by nutation, causes it alternately to approach 
to and recede from the stars, and thus to change their declinations. 
The solar nutation, depending on the position of the sun with re- 
spect to the equinoxes, passes through all its variations annually ; 
but the' lunar nutation depending on the position of the moon with 
respect to her nodes, varies through a period of about 18 years. 

ABERRATION. 

195. ABERRATION is an apparent change of place in the stars, 
occasioned by the joint effects of the motion of the earth in its orbit, 
and the progressive motion of light. 

Let EE' (Fig. 43,) represent a part of the earth's orbit, and SE 
a ray of light from the star S. Take EC and EA proportional 




MOTION OF THE APSIDES. 109 

tc the velocity of each respectively ; com- 

ple<;e the parallelogram, and draw the diagonal 

EB. Sijice an object always appears in the 

direction in which a ray of light coming from 

it, meets the eye, the combination of the two 

motions produces an impression on the eye 

exactly similar to that which would have been 

produced if the eye had remained at rest in 

the point E, and the particle of light had come 

down to it in the direction S'E ; the star, 

therefore, whose place is at S, will appear to 

the spectator at E to be situated at S'. The 

difference between its true and its apparent place, that is, the 

angle SES' is the aberration, the magnitude of which is obtained 

from the known ratio of EA to EC, or the velocity of light to that 

of the earth in its orbit. 

The velocity of light is 192,000 miles per second, while that of 
the earth in its orbit is about 19 miles per second. Represent- 
ing the velocity of light by the line EA, and that of the earth by 
AB, then, 

192,000 : 19: :Rad. : tan. 20."5=the angle at E, which is the 
amount of aberration when the direction of the ray of light is per- 
pendicular to the earth's motion. 

The effect of aberration upon the places of the fixed stars is to 
carry their apparent places a little forward of their real places in 
the direction of the earth's motion. The effect upon each particu- 
lar star will be to make it describe a small ellipse in the heavens, 
having for its center the point in which the star would be seen if 
the earth were at rest. 



MOTION OF THE APSIDES. 

196. The two points of the ecliptic where the earth is at the 
greatest and least distances from the sun respectively, do not 
always maintain the same places among the signs, but gradually 
shift their positions from west to east. If we accurately observe 
the place among the stars, where the earth is at the time of its 
perihelion the present year, we shall find that it will not be pre- 



110 THE SUN. 

cisely at that point the next year when it arrives at its perihelion, 
but about 12" (ll."66) to the east of it. And since the equinox 
itself, from which longitude is reckoned, moves in the opposite 
direction 50." 1 annually, the longitude of the perihelion increases 
every year 61. "76, or a little more than one minute. This fact 
is expressed by saying that the line of the apsides of the earth's 
orbit has a slow motion from west to east. It completes one entire 
revolution in its own plane in about 100,000 years (111,149.) 

The mean longitude of the perihelion at the commencement of 
the present century was 99 30' 5", and of course in the ninth 
degree of Cancer, a little past the winter solstice. In the year 
1248, the perihelion was at the place of this solstice ; and since the 
increase of longitude is 61. "76 a year, hence, 

61. "76 : 1 : : 90 : 5246 =the time occupied in passing from the 
first of Aries to the solstice. Hence, 52461248=3998, which is 
the time before the Christian era, when the perigee was at the 
first of Aries. But this differs only 6 years from the time of the 
creation of the world, which is fixed by chronologists at 4004 
years A. C. At the period of the creation, therefore, the line of 
the apsides of the earth's orbit, coincided with the line of the 
equinoxes. 

197. The angular distance of a body from its aphelion is called 
its Anomaly ; and the interval between the sun's passing the point 
of the ecliptic corresponding to the earth's aphelion, and return- 
ing to the same point again, is called the anomalistic year. This 
period must be a little longer than the sidereal year, since, in order 
to complete the anomalistic revolution^ the sun must traverse an 
arc of 11. "66 in addition to 360. 

Now 360 : 365.256 : : 11. "66 : 4m. 44s. 

198. Since the points of the annual orbit, where the sun is at 
the greatest and least distances from the earth, change their posi- 
tion with respect to the solstices, a slow change is occasioned in 
the duration of the respective seasons. For, let the perihelion 
correspond to the place of the winter solstice, as was the case in 
the year 1248 ; then as the sun moves more rapidly in that part 
of his orbit, the winter months will be shorter than the summer. 



f 
MEAN AND TRUE PLACES OP THE SUN. Ill 

But, again, let the perihelion be at the summer solstice, as it will 
be in the year 6485* ; then the sun will move most rapidly 
through thfc summer months, and the winters will be longer than 
the summers. At present the perihelion is so near the winter 
solstice, that, the year being divided into summer and winter by 
the equinoxes, the six winter months are passed over between seven 
and eight days sooner than the summer months. 

MEAN AND TRUE PLACES OP THE SUN. 

199. The Mean Motion of any body revolving in an orbit, is 
that which it would have if, in the same time, it revolved uniformly 
in a circle. 

In surveying an irregular field, it is common first to strike out 
some regular figure, as a square or a parallelogram, by running 
long lines, and disregarding many small irregularities in the boun- 
daries of the field. By this process, we obtain an approximation 
to the contents of the field, although we have perhaps thrown out 
several small portions which belong to it, and included a number 
of others which do not belong to it. These being separately esti- 
mated and added to or substracted from our first computation, we 
obtain the true area of the field. In a similar manner, we proceed 
in finding the place of a heavenly body, which moves in an orbit 
more or less irregular. Thus we estimate the sun's distance from 
the vernal equinox for every day of the year at noon, on the 
supposition that he moves uniformly in a circular orbit : this is 
the sun's mean longitude. We then apply to this result various 
corrections for the irregularity of the sun's motions, and thus ob- 
tain the true longitude. 

200. The corrections applied to the mean motions of a heav- 
enly body, in order to obtain its true place, are called Equations. 
Thus the elliptical form of the earth's orbit, the precession of the 
equinoxes, and the nutation of the earth's axis, severally affect 
the place of the sun in his apparent orbit, for which equations are 
applied. In a collection of Astronomical Tables, a large part of 



* Biot. 



112 THE SUN. 

the whole are devoted to this object. They give us the amount 
of the corrections to be applied under all the circumstances and 
constantly varying relations in which the sun, moon, and earth 
are situated with respect to each other. The angular distance of 
the earth or any planet from its aphelion, on the supposition that 
it moves uniformly in a circle, is called its Mean Anomaly : its 
actual distance at the same moment in its orbit is called its True 
Anomaly* 

Thus in figure 44, let AEB represent the orbit of the earth 
having the sun in one of the foci at S. Upon AB describe the 
circle AMB. Let E be the place of the earth in its orbit, and M 
the corresponding place in the circle ; then the angle MCA is the 
mean, and ESA the true anomaly. The difference between the 

Fig. 44. 

M 




mean and true anomaly, MCA ESA, is called the the Equation of 
the Center, being that correction which depends on the elliptical 
form of the orbit, or on the distance of the center of attraction 
from the center of the figure, that is, on the eccentricity of the 
orbit. It is much the greatest of all the corrections used in finding 
the sun's true longitude, amounting, at its maximum, to nearly two 
degrees (1 55' 26."8.) 

* In some astronomical treatises, the anomaly is reckoned from the perihelion. 



CHAPTER V. 

OF THE MOON LUNAR GEOGRAPHY* PHASES OF THE MOON HER 

REVOLUTIONS. 

201. NEXT to the Sun, the Moon naturally claims our attention. 
The Moon is an attendant or satellite to the earth, around which 

she revolves at the distance of nearly 240,000 miles. Her mean / 
horizontal parallax being 57' 09",f consequently, sin. 57' 09" : 
semi-diameter of the earth (3956.2) : : rad. : 238,545. (Art. 87.) 

The moon's apparent diameter is 31' 7", and her real diameter 2 ~i 3 
2160 miles. For, 

Rad. : 238,545 :: sin. 15' 33|" : 1079.6. = moon's semi-diame- 
ter. (See Fig. 26, p. 71.) 

And, since spheres are as the cubes of the diameters, the vol- 
ume of the moon is T V that of the earth. Her density is nearly 
f (.615) the density of the earth, and her mass (=JjX.6I5) is 
about aV- 

202. The moon shines by reflected light borrowed from the 
sun, and when full, exhibits a disk of silvery brightness, diversi- 
fied by extensive portions partially shaded. By the aid of the 
telescope, we see undoubted signs of a varied surface, composed 
of extensive tracts of level country, and numerous mountains and 
valleys. 

203. The line which separates the enlightened from the dark 
portions of the moon's disk, is called the Terminator. (See Fig. 2. 
Frontispiece.) As the terminator traverses the disk from new to 
full rnoon, it appears through the telescope exceedingly broken in 



* Selenography is a word more appropriate to a description of the moon, but is not 
perhaps sufficiently familiarized by use. 
t Daily's Astronomical Tables. 

15 



114 THE MOON. 

some parts, but smooth in others, indicating that some portions of the 
lunar surface are uneven while others are level. The broken re- 
gions appear brighter than the smooth tracts. The latter have 
been taken for seas, but it is supposed with more probability that 
they are extensive plains, since they are still too uneven for the 
perfect level assumed by bodies of water. That there are moun- 
tains in the moon, is known by several distinct indications. First, 
when the moon is increasing, certain spots are illuminated sooner 
than the neighboring places, appearing like bright points beyond 
the terminator, within the dark part of the disk. (See Fig. 2. 
Frontispiece) Secondly, after the terminator has passed over 
them, they project shadows upon the illuminated part of the disk, 
always opposite to the sun, corresponding in shape to the form of 
the mountain, and undergoing changes in length from night to 
night, according as the sun shines upon that part of the moon 
more or less obliquely. Many individual mountains rise to a great 
height in the midst of plains, and there are several very remarka- 
ble mountainous groups, extending from a common center in long 
chains. 

204. That there are also valleys in the moon, is equally evident. 
The valleys are known to be truly such, particularly by the man- 
ner in which the light of the sun falls upon them, illuminating the 
part opposite to the sun while the part adjacent is dark, as is the 
case when the light of a lamp shines obliquely into a china cup. 
These valleys are often remarkably regular, and some of them 
almost perfect circles. In several instances, a circular chain of 
mountains surrounds an extensive valley, which appears nearly 
level, except that a sharp mountain sometimes rises from the cen- 
ter. The best time for observing these appearances is near the 
first quarter of the moon, when 'half the disk is enlightened ;* 
but in studying the lunar geography, it is expedient to observe the 
moon ever) evening from new to full, or rather through her en- 
tire series of changes. 



* It is earnestly recommended to the student of astronomy, to examine the moon re- 
peatedly with the best telescope he can command, using low powers at first, for the 
sake of a better light. 



LUNAR GEOGRAPHY. 115 

205. The various places on the moon's disk have received ap- 
propriate names. The dusky regions, being formerly supposed to 
be seas, were named accordingly ; and other remarkable places 
have each two names, one derived from some well known spot on 
the earth, and the other from some distinguished personage. Thus 
the same bright spot on the surface of the moon is called Mount 
Sinai or Tycho, and another Mount Etna or Copernicus. The 
names of individuals, however, are more used than the others. 
The frontispiece exhibits the telescopic appearance of the full 
moon. A few of the most remarkable points have the following 
names, corresponding to the numbers and letters on the map. (See 
Frontispiece.) 

1. Tycho, A. Mare Humorum, 

2. Kepler, B. Mare Nubium, 

3. Copernicus, C. Mare Imbrium, 

4. Aristarchus, D. Mare Nectaris, 

5. Helicon, E. Mare Tranquilitatis, 

6. Eratosthenes, F. Mare Serenitatis, 

7. Plato, G. Mare Fecunditatis, 

8. Archimedes, H. Mare Crisium. 

9. Eudoxus, 
10. Aristotle, 

206. The method of estimating the height of lunar mountains is 
as follows. 

Let ABO (Fig. 45,) be the illuminated hemisphere of the moon, 
SO a solar ray touching the moon in O, a point in the circle which 
separates the enlightened from the dark part of the moon. All the 
part ODA will be in darkness ; but if this part contains a moun- 
tain MF, so elevated that its summit M reaches to the solar ray 
SOM, the point M will be enlightened. Let E be the place of the 
observer on the earth, the moon being at any elongation from the 
sun, as measured by the angle EOS. Draw the lines EM, EO, 
and CM, C being the center of the moon ; and let FM be the 
height of the mountain. Draw ON perpendicular to EM. The 
line EO being known, and the angle OEM being measured with a 
micrometer, the value of ON, the projection of the lime OM, be- 



116 




comes known. Now OM 



ON 



^.TVT 5 an d since OEN is a very 
cos. MON 

small angle, EON may be considered as a right angle ; conse- 

ON 



quently, MON=MOE-90. 
ON ON 



Therefore OM=- 



cos. (MOE-90) 
That is, the distance between the summit 



sin. MOE sin. EOS' 
of the mountain and the illuminated part of the moon's disk, is 
equal to the projected distance as measured by the micrometer, 
divided by the sine of the moon's elongation from the sun. 

Suppose the distance OM=7zCO, where n represents the frac 
tion the part OM is of CO as determined by observation. Then, 
CM 2 =CO 2 +OM 2 =CO 2 +n 2 CO 3 =CO 2 (1-f-n 2 ) .-. CM-CO (l+n 2 )| 

/.CM-CO or FM=CO (x/I+n 2 -!) = CO, neglecting the 

higher powers of w, which would be of too little value to be worth 
taking into the account. The value of n has been found in one 
case equal to T V, which gives the height of the mountain equal to 
^I T the semi-diameter of the moon, that is, 3} miles. 

When the moon is exactly at quadrature, then EOM becomes a 
right angle, and the value of OM is obtained directly from actual 
measurement ; and having CO and OM, we easily obtain CM and 
of course FM. 



LUNAR GEOGRAPHY. 117 

207. Schroeter, a German astronomer, estimated the heights of 
the lunar mountains by observations on their shadows. He made 
them in some cases as high as ^ of the semi-diameter of the 
moon, that is, about 5 miles. The same astronomer also estimates 
the depths of some of the lunar valleys at more than four miles. 
Hence it is inferred that the moon's surface is more broken and 
irregular than that of the earth, its mountains being higher and its 
valleys deeper in proportion to the size of the moon than those of 
the earth. 

i 

208. Dr. Herschel is supposed also to have obtained decisive 
evidence of the existence of volcanoes in the moon, not only 
from the light afforded by their fires, but also from the formation 
of new mountains by the accumulation of matter where fires had 
been seen to exist, and which remained after the fires were extinct. 

209. Some indications of an atmosphere about the moon have 
been obtained, the most decisive of which are derived from ap- 
pearances of twilight, a phenomenon that implies the presence 
of an atmosphere. Similar indications have been detected, it is 
supposed, in eclipses of the sun, denoting a transparent refracting 
medium encompassing the moon. The lunar atmosphere, how- 
ever, if any exists, is very inconsiderable in extent and density 
compared with that of the earth.* 

210. The improbability of our ever identifying artificial struc- 
tures in the moon may be inferred from the fact that a line one 
mile in length in the moon subtends an angle at the eye of only 
about one second. If, therefore, works of art were to have a suf- 
ficient horizontal extent to become visible, they can hardly be sup- 
posed to attain the necessary elevation, when we reflect that the 
height of the great pyramid of Egypt is less than the sixth part of 
a mile. 

* See Ed. Encyc. 11.598. 



118 THE MOON. 

PHASES OF THE MOON. 

211. The changes -of the moon, commonly called her Phases, 
arise from different portions of her illuminated side being turned 
towards the earth at different times. When the moon is first 
seen after the setting sun, her form is that of a bright crescent, 
on the side of the disk next to the sun, while the other portions 
of the disk shine with a feeble light, reflected to the moon from 
the earth. Every night we observe the moon to be further and 
further eastward of the sun, and at the same time the crescent 
enlarges, until, when the moon has reached an elongation from 
the sun of 90, half her visible disk is enlightened, and she is 
said to be in her first quarter. The terminator, or line which 
separates the illuminated from the dark part of the moon, is con- 
vex towards the sun from the new moon to the first quarter, and 
the moon is said to be horned. The extremities of the crescent 
are called cusps. At the first quarter, the terminator becomes a 
straight line, coinciding with a diameter of the disk ; but after 
passing this point, the terminator becomes concave towards the 
sun, bounding that side of the moon by an elliptical curve, when 
the moon is said to be gibbous. When the moon arrives at the 
distance of 180 from the sun, the entire circle is illuminated, 
and the moon is full. She is then in opposition to the sun, rising 
about the time the sun sets. For a week after the full, the moon 
appears gibbous again, until, having arrived within 90 of the sun, 
she resumes the same form as at the first quarter, being then at 
her third quarter. From this time until new moon, she exhibits 
again the form of a crescent before the rising sun, until approach- 
ing her conjunction with the sun, her narrow thread of light is lost 
in the solar blaze ; and finally, at the moment of passing the sun, 
the dark side is wholly turned towards us and for some time we 
lose sight of the moon. 

The two points in the orbit corresponding to new and full moon 
respectively, are called by the common name of syzygies ; those 
which are 90 from the sun are called quadratures; and the 
points half way between the syzygies and quadratures are called 
octants. The circle which divides the enlightened from the unen- 
Mghtened hemisphere of the moon, is called the circle of illumma 



PHASES. 



119 



tion ; that which divides the hemisphere that is turned towards 
us from the hemisphere that is turned from us, is called the circle 
of the disk. 

212. As the moon is an opake body of a spherical figure, and 
borrows her light from tfte sun, it is obvious that that half only 
which is towards the sun can be illuminated. More or less of 
this side is turned towards the earth, according as tne moon is at 
a greater or less elongation from the sun. The reason of the dif- 
ferent phases will be best understood from a diagram. Therefore 
let T (Fig. 46,) represent the earth, and S the sun. Let A, B, C, 
fec., be successive positions of the moon. At A the entire dark 

Fig. 46. 




side of the moon being turned towards the earth, the disk would 
be wholly invisible. At B, the circle of the disk cuts off a small 
part of the enlightened hemisphere, which appears in the heavens 
at 6, under the form of a crescent. At C, the first quarter, the 
circle of the disk cuts off half the enlightened hemisphere, and the 
moon appears dichotomized at c. In like manner it will be seen 
that the appearances presented at D, E, F, &c., must be those 
represented at d, e,f. 



REVOLUTIONS OF THE MOON. 



213. The moon revolves around the earth from west to east, 
making the entire circuit of the heavens in about 27 days. 



~\ 



120 THE MOON. 

The precise law of the moon's motions in her revolution around 
the earth, is ascertained, as in the case of the sun, (Art. 155,) by 
daily observations on her meridian altitude and right ascension. 
Thence are deduced by calculation her latitude and longitude, 
from which we find, that the moon describes on the celestial 
sphere a great circle of which the eaMh^the center. 

The period of the moon's revolution from any point in the 
heavens round to the same point again, is called a month. A 
sidereal month is the time of the moon's passing from any star, 
until it returns to the same star again. A synodical month* is 
the time from one conjunction or new moon to another. The 
synodical month is about 29^ days, or more exactly, 29d. 12h. 
44m. 2 S .8=29.53 days. The sidereal month is about two days 
shorter, being 27d. 7h. 43m. 1 l s .5=27.32 days. As the sun and 
moon are both revolving in the same direction, and the sun is 
moving nearly; a degree a day, during the 27 days of the moon's 
revolution, the sun must have moved 27. Now since the moon 
passes over 360 in 27.32 days, her daily motion must be 13 17'. 
It must therefore evidently take about two days for the moon to 
overtake the sun. The difference between these two periods 
may, however, be determined with great exactness. The mid- 
dle of an eclipse of the sun marks the exact moment of conjunc- 
tion or new moon ; and by dividing the interval between any 
two solar eclipses by the number of revolutions of the moon, or 
lunations, we obtain the precise period of the synodical month. 
Suppose, for example, two eclipses occur at an interval of 1,000 
lunations ; then the whole number of days and parts of a day 
that compose the interval divided by 1,000 will give the exact 
time of one lunation.f The time of the synodical month being 
ascertained, the exact period of the sidereal month may be derived 
from it. For the arc which the moon describes in order to come 
into conjunction with the sun, exceeds 360 by the space which 



* yvv and o<5of, implying that the two bodies come together. 

t It might at first view seem necessary to know the period of one lunation before 
we could know the number of lunations in any given interval. This period is known 
very nearly from the interval between one new moon and anothe* 



the sun has passed over since the preceding conjunction, that is, 
in 29.53 days. Therefore, 

365.24 : 360 :: 29.53 : 29. 1 arc which the moon must de- 
scribe more than 360 in order to overtake the sun. Hence, 

13 17' : Id.: : 29. 1 : 2.2 Id. ^difference between the sidereal 
and synodical months ; and 29.53 2.21=27.32, the time of the 
sidereal revolution. 

214. The moori 's orbit is inclined to the ecliptic in an angle of 
about 5 (5 8' 48"). It crosses the ecliptic in two opposite points / 
called her nodes. The amount of inclination is ascertained by 
observations on the moon s latitude when at a maximum, that 
being of course the greatest distance from the ecliptic, and there- 
fore equal to the inclination of the two circles. 

215. The moon, at the same age, crosses the meridian at differ- 
ent altitudes at different seasons of the year. The full moon, for 
example, will appear much further in the south when on the meri- 
dian at one period of the year than at another. This is owing to 
the fact that the moon's path is differently situated with respect to 
the horizon, at a given time of night at different seasons of the 
year. By taking the ecliptic on an artificial globe to represent 
the moon's path, (which is always near it, Art. 214,) and recollect- 
ing that the new moon is seen in the same part of the heavens 
with the sun, and the full moon in the opposite part of the heavens 
from the sun, we shall readily see that in the winter the new 
moons must run low because the sun does, and for a similar rea- 
son the full moons must run high. It is equally apparent that, in 
summer, when the sun runs high, the new moons must cross the 
meridian at a high, and the full moons at a low altitude. This 
arrangement gives us a great advantage in respect to the amount 
of light received from the moon ; since the full moon is longest 
above the horizon during the long nights of winter, when her pre- 
sence is most needed. This circumstance is especially favorable to 
the inhabitants of the polar regions, the moon, when full, travers- 
ing that part of her orbit which lies north of the equator, and of 
course above the horizon of the north pole, and traversing the por- 
tion that lies south of the equator, and below the polar horizon, 

16 



122 THE MOON. 

when new. During the polar winter, therefore, the moon, from 
the first to the last quarter, is commonly above the horizon, wnile 
the sun is absent ; whereas, during summer, while the sun is pre- 
sent, the moon is above the horizon while describing her first and 
last quadrants. 

216. About the time of the autumnal equinox, the moon when 
near the full, rises about sunset for a number of nights in succes- 
sion ; and as this is, in England, the period of harvest, the phe- 
nomenon is called the Harvest Moon. To understand the reason 
of this, since the moon is never far from the ecliptic, we will 
suppose her progress to be in the ecliptic. If the moon moved 
in the equator, then, since this great circle is at right angles to 
the axis of the earth, all parts of it, as the earth revolves, cut the 
horizon at the same constant angle. But the moon's orbit, or 
the ecliptic, which is here taken to represent it, being oblique 
to the equator, cuts the horizon at different angles in different 
parts, as will easily be seen by reference to an artificial globe. 
When the first of Aries, or vernal equinox, is in the eastern hori- 
zon, it will be seen that the ecliptic, (and consequently the moon's 
orbit,) makes its least angle with the horizon. Now at the au- 
tumnal equinox, the sun being in Libra, the moon at the full is in 
Aries, and rises when the sun sets. On the following evening, 
although she has advanced in her orbit about 13, (Art. 213,) yet 
her progress being oblique to the horizon, and at a small angle 
with it, she will be found at this time but a little way below the 
horizon, compared with the point where she was at sunset the 
preceding evening. She therefore rises but little later, and so 
for a week only a little later each evening than she did the pre- 
ceding night. 

217. The moon is about -$\ nearer to us when near the zenith 
than when in the horizon. 

The horizontal distance CD (Fig. 47,) is nearly equal to AD= 
AD', which is greater than CD' by AC, the semi-diameter of the 
earth = ? V tne distance of the moon. Accordingly, the apparent 
diameter of the moon, when actually measured, is about 30" 
(which equals about T V of the whole) greater when in the zenith 



REVOLUTIONS. 123 

than in the horizon. The apparent enlargement of the full moon 
when rising, & owing to the same causes as that of the sun, as ex- 
plained in article 96. 

Fig. 47. 




218. The moon turns on its axis in the same time in which it 
revolves around the earth. 

This is known by the moon's always keeping nearly the same 
face towards us, as is indicated by the telescope, which could not 
happen unless her revolution on her axis kept pace with her mo- 
tion in her orbi't. Thus, it" will be seen by inspecting figure 31, 
that the earth turns different faces towards the sun at different 
times ; and if a ball having one hemisphere white and the other 
black be carried around a lamp, it will easily be seen that it can- 
not present the same face constantly towards the lamp unless it 
turns once on its axis while performing its revolution. The same 
thing will be observed when a man walks aiound a tree, keeping 
his face constantly towards it. Since however the motion of the 
moon on its axis is uniform, while the motion in its orbit is une- 
qual, the moon does in fact reveal to us a little sometimes of one 
side and sometimes of the other. Thus when the ball above 
mentioned is placed before the eye with its light side towards us, 
or carrying it round, if it is moved faster than it is turned on its 
axis, a portion of the dark hemisphere is brought into view on 
one side ; or if it is moved forward slower than it is turned on 
its axis, a portion of the dark hemisphere comes into view on the 
other side. 

219. These appearances are called the moon's librations in lon- 
gitude. The moon has also a libration in latitude, so called, be- 
cause in one part of her revelation, more of the region around one 



124 THE MOON. 

of the poles comes into view, and in another part of the revolu- 
tion, more of the region around the other pole ; which gives the ap- 
pearance of a tilting motion to the moon's axis. This has nearly the 
same cause with that which occasions our change of seasons. The 
moon's axis being inclined to that of the ecliptic, about 1| degrees, 
(1 30' 10".8,) and always remaining parallel to itself, the circle 
which divides the visible from the invisible part of the' moon, will 
pass in such a way as to throw sometimes more of one pole into 
view and sometimes more of the other, as would be the case with 
the earth if seen from the sun. (See Fig. 31.) 

The moon exhibits another phenomenon of this kind called 
her diurnal libration, depending on the daily rotation of the 
spectator. She turns the same face towards the center of the 
earth only, whereas we view her from the surface. When she is 
on the meridian, we see her disk nearly as though we viewed it 
from the center of the earth, and hence in this situation it is sub- 
ject to little change ; but when near the horizon, our circle of 
vision takes in more of the upper limb than would be presented 
to a spectator at the center of the earth. Hence, from this cause, 
we see a portion of one limb while the moon is rising, which is 
gradually lost sight of, and we see a portion of the opposite limb 
as the moon declines towards the west. It will be remarked that 
neither of the foregoing changes implies any actual motion in the 
moon, but that each arises from a change of position in the spec- 
tator relative to the moon. 

220. An inhabitant of the moon would have but one day and 
one night during the whole lunar month of 29 days. One of 
its days, therefore, is equal to nearly 15 of ours. So protracted 
an exposure to the sun's rays, especially in the equatorial regions 
of the moon, must occasion an excessive accumulation of heat ; 
and so long an absence of the sun must occasion a corresponding 
degree of cold. Each day would be a wearisome summer ; each 
night a severe winter.* A spectator on the side of the moon 
which is opposite to us would never see the earth ; but one on the 
side next to us would see the earth presenting a gradual succession 

* Francceur, Uranog. p. 91. 



REVOLUTIONS. 125 

of changes during his long night of 360 hours. Soon after the 
earth's conjunction with the sun, he would have the light of the 
earth reflected to him, presenting at first a crescent, but enlarging, 
as the earth approaches its opposition, to a great orb, 13 times as 
large as the full moon appears to us, and affording nearly 13 times 
as much light. Our seas, our plains, our mountains, our volcanoes, 
and our clouds, would produce very diversified appearances, as 
would the various parts of the earth brought successively into 
view by its diurnal rotation. The earth while in view to an in- 
habitant of the moon, would remain immovably fixed in the same 
part of the heavens. For being unconscious of his own motion 
around the earth, as we are of our motion around the sun, the 
earth would seem to revolve around his own planet from west to 
east ; but, meanwhile, his rotation along with the moon on her 
axis, would cause the earth to have an apparent motion westward 
at the same rate. The two motions, therefore, would exactly 
balance each other, and the earth would appear all the while at 
rest. The earth is full to the moon when the latter is new to us ; 
and universally the two phases are complementary to each other.* 

221. It has been observed already, (Art. 214,) that the moon's 
orbit crosses the ecliptic in two opposite points called the nodes. 
That which the moon crosses fVom south to north, is called the 
ascending node ; that which the moon crosses from north to south, 
the descending node. 

From the manner in which the figure representing the earth's 
orbit and that of the moon, is commonly drawn, the learner is 
sometimes puzzled to see how the orbit of the moon can cut the 
ecliptic in two points directly opposite to each other. But he must 
reflect that the lunar orbit cuts the plane of the ecliptic and not 
the earth's path in that plane, although these respective points are 
projected upon that path in the heavens. 

222. We have thus far contemplated the revolution of the moon 
around the earth as though the earth were at rest. But, in order 
to have just ideas respecting the moon's motions, we must recol- 
lect that the moon likewise revolves along with the earth around 

* Francoeur p. 92. 



126 THE MOON. 

the sun. It is sometimes said that the earth carries the moon 
along with her in her annual revolution. This language may 
convey an erroneous idea ; for the moon, as well as the earth, 
revolves around the sun under the influence of two forces, and 
would continue her motion around the sun, were the earth re 
moved out of the way. Indeed, the moon is attracted towards 
the sun 2} times more than towards the earth,* and would aban- 
don the earth wej;e not the latter also carried along with her by 
the same forces. So far as the sun acts equally on both bodies, 
their motion with respect to each other would not be disturbed. 
Because the gravity of the moon towards the sun is found to be 
greater, at the conjunction, than her gravity towards the earth, 
some have apprehended that, if the doctrine of universal gravi- 
tation is true, the moon ought necessarily to abandon the earth. 
In order to understand the reason why it does not do thus we 
must reflect, that when a body is revolving in its orbit under the 
action of the projectile force and gravity, whatever diminishes 
the force of gravity while that of projection remains the same, 
causes the body to recede, from the center; and whatever in- 
creases the amount of gravity carries the body towards the center 
Now? when the moon is in conjunction, her gravity towards the 
earth acts in opposition to that towards the sun, while her velocity 
remains too great to carry her*, with what force remains, in a 
circle about the sun, and she therefore recedes from the sun, and 
commences her revolution around the earth. On arriving at the 
opposition, the gravity of the earth conspires with that of the sun, 
and the moon's projectile force being less than that required to 
make her revolve in a circular orbit, when attracted towards the 
sun by the sum of these forces, she accordingly begins to approach 
the sun and descends again to the conjunction.! 



* It is shown by writers on Mechanics, that the forces with which bodies revolving 
in circular orbits tend towards their centers, are as the radii of their orbits divided 
by the squares of their periodical times. Hence, supposing the orbits of the earth and 
the moon to be circular, (and their slight eccentricity will not much affect Ihe re- 
sult,) we have 

400 1 
C C*> 9 9 . 1 

' (365.25)* : (27.32) a * ' 
t M'Laurin's Discoveries of Newton, B. iv, ch. 5. 



LUNAR IRREGULARITIES. 127 

223. The attraction of the sun, however, being every where 
greater than that of the earth, the actual path of the moon around 
the sun is every where concave towards the latter. Still the el- 
liptical path of the moon around the earth, is to be conceived of 
in the same way as though both bodies were at rest with respect 
to the sun. Thus, while a steamboat is passing swiftly around an 
island, and a man is walking slowly around a post in the cabin, 
the line which he describes in space between the forward motion 
of the boat and his circular motion around the post, may be every 
where concave towards the island, while his path around the post 
will still be the same as though both were at rest. A nail in the 
rim of a coach wheel, will turn around the axis of the wheel, when 
the coach has a forward motion in the same manner as when the 
coach is at rest, although the line actually described by the nail 
will be the resultant of both motions, and very different from 
either. 



CHAPTER VI. 

LUNAR IRREGULARITIES. 

224. WE have hitherto regarded the moon as describing a great 
circle on the face of the sky, such being the visible orbit as seen 
by projection. But, on more exact investigation, it is found that 
her orbit is not a circle, and that her motions are subject to very 
numerous irregularities. These will be best understood in con- 
nection with the causes on which they depend. The law of uni- 
versal gravitation has been applied with wonderful success to their 
investigation, and its results have conspired with those of long 
continued observation, to furnish the means of ascertaining with 
great exactness the place of the moon in the heavens at any given 
instant of time, past or future, and thus to enable astronomers to 
determine longitudes, to calculate eclipses, and to solve various 
other problems of the highest interest. A complete understand- 
ing of all the irregularities of the moon's motions, must be sought 



128 THE MOON. 

for in more extensive treatises of astronomy than the present ; but 
some general acquaintance with the subject, clear and intelligible 
as far as it goes, may be acquired by first gaining a distinct idea 
of the mutual actions of the sun, the moon, and the earth. 

225. The irregularities of the moon's motions, are due chiefly to 
the disturbing influence of the sun, which operates in two ways ; flrst, 
by acting unequally on the earth and moon, and, secondly, by acting 
obliquely on the moon, on account of the inclination of her orbit to 
the ecliptic.* 

If the sun acted equally on the earth and moon, and always in 
parallel lines, this action would serve only to restrain them in their 
annual motions round the sun, and would not affect their actions 
on each other, or their motions about their common center of 
gravity. In that case, if they were allowed to fall directly to- 
wards the sun, they would fall equally, and their respective situa- 
tions would not be affected by their descending equally towards 
it. We might then conceive them as in a plane, every part of 
which being equally acted on by the sun, the whole plane would 
descend towards the sun, but the respective motions of the earth 
and the moon in this plane, would be the same as if it were qui- 
escent. Supposing then this plane and all in it, to have an annual 
motion imprinted on it, it would move regularly round the sun. 
while the earth and moon would move in it with respect to each 
other, as if the plane were at rest, without any irregularities. 
But because the moon is nearer the sun in one half of her orbit 
than the earth is, and in the other half of her orbit is at a greater 
distance than the earth from the sun, while the power of gravity 
is always greater at a less distance ; it follows, that in one half of 
her orbit the moon is more attracted than the earth towards the 
sun, and in the other half less attracted than the earth. The ex- 
cess of the attraction, in the first case, and the defect in the second, 
constitutes a disturbing force, to which we may add another, 
namely, that arising from the oblique action of the solar force, 
since this action is not directed in parallel lines, but in lines thai 
meet in the center of the sun. 

* M'Laurin's Discoveries of Newton, B. iv, ch. 4. La Place's Syst. du Monde, 
B. iv, ch. 5. 



LUNAR IRREGULARITIES. 129 

226. To see the effects of this process, let us suppose that the 
projectile motions of the earth and moon were destroyed, and 
that they were allowed to fall freely towards the sun. If the 
moon was in conjunction with the sun, or in that part of her orbit 
which is nearest to him, the moon would be more attracted than 
the earth, and fall with greater velocity towards the sun ; so that 
the distance of the moon from the earth would be increased in the 
fall. If the moon was in opposition, or in the part of her orbit 
which is furthest from the sun, she would be less attracted than 
the earth by the sun, and would fall with a less velocity towards 
the sun, and would be left behind ; so that the distance of the 
moon from the earth would be increased in this case also. If the 
moon was in one of the quarters, then the earth and moon being 
both attracted towards the center of the sun, they would both de- 
scend directly towards that center, and by approaching it, they 
would necessarily at the same time approach each other, and in 
this case their distance from each other would be diminished. 
Now whenever the action of the sun would increase their distance, 
if they were allowed to fall towards the sun, then the sun's action, 
by endeavoring to separate them, diminishes their gravity to each 
other ; whenever the sun's action would diminish the distance, then 
it increases their mutual gravitation. Hence, in the conjunction 
and opposition, that is, in the syzygies, their gravity towards each 
other is diminished by the action of the sun, while in the quadra- 
tures it is increased. But it must be remembered that it is not 
the total action of the sun on them that disturbs their motions, 
but only that part of it which tends at one time to separate them, 
and at another time to bring them nearer together. The other 
and far greater part, has no other effect than to retain them in 
their annual course around the sun. 

227. Suppose the moon setting out from the quarter that pre- 
cedes the conjunction with a velocity that would make her de- 
scribe an exact circle round the earth, if the sun's action had no 
effect on her : since her gravity is increased by that action, she must 
descend towards the earth and move within that circle. Her or- 
bit then would be more curved than it otherwise would have been; 
because the addition to her gravity will make her fall further at 

17 



130 



THE MOON. 



the end of an arc below the tangent drawn at the other end of it 
Her motion will be thus accelerated, and it will continue to be 
accelerated until she arrives at the ensuing conjunction, because 
the direction of the sun's action upon her, during that time, makes 
an acute angle with the direction of her motion. (See Fig. 41.) 
At the conjunction, her gravity towards the earth being diminished 
by the action of the sun; her orbit will then be less curved, and 
she will be carried further from the earth as she moves to the next 
quarter ; and because the action of the sun makes there an obtuse 
angle with the direction of her motion, she will be retarded in the 
same degree as she was accelerated before. 

228. After this general explanation of the mode in which the 
sun acts as a disturbing force on the motions of the moon, the 
learner will be prepared to understand the mathematical develop- 
ment of the same doctrine. 

Therefore, let ADBC (Fig. 48,) be the orbit, nearly circular, in 
which the moon M revolves in the direction CADB, round the 
earth E. Let S be the sun, and let 
SE the radius of the earth's orbit, 
be taken to represent the force with 
which the earth gravitates to the sun. 

Then (Art. 180,) ~: JL : :SE : 

SE 3 

*|2 .= the force by which the sun 

draws the moon in the direction 
MS. Take MG=^^r, and let the 



parallelogram KF be described, 
having MG for its diagonal, and 
having its sides parallel to EM and 
ES. The force MG may be re- 
solved into two, MF and MK, of 
which MF, directed towards E, the 
center of the earth, increases the 
gravity of the moon to the earth, and does not hinder the areas 
described by the radius vector from being proportional to the 




LUNAR IRREGULARITIES. 1 3 1 

times. The other force MK draws the moon in the direction of 
the line joining the centers of the sun and earth. It is, however 
only the excess of this force, above the force represented by SE 
or that which draws the earth to the sun, which disturbs the rela- 
tive position of the moon and earth. This is evident, for if KM 
were just equal to ES, no disturbance of the moon relative to the 
earth could arise from it. If then ES be taken from MK, the dif- 
ference HK is the whole force in the direction parallel to SE, by 
which the sun disturbs the relative position of the moon and earth. 
Now, if in MK, MN be taken equal to HK, and if NO be drawn 
perpendicular to the radius vector EM produced, the force MN 
may be resolved into two, MO and ON, the first lessening the 
gravity of the moon to the earth ; and the .second, being parallel 
to the tangent of the moon's orbit in M, accelerates the moon's 
motion from C to A, and retards it from A to D, and so alternately 
in the other two quadrants. Thus the whole solar force directed 
to the center of the earth, is composed of the two parts MF and 
MO, which are sometimes opposed to one another, but which 
never affect the uniform description of the areas about E. Near 
the quadratures the force MO vanishes, and the force MF, which 
increases the gravity of the moon to the earth, coincides with CE 
or DE. As the moon approaches the conjunction at A, the force 
MO prevails over MF, and lessens the gravity of the moon to the 
earth. In the opposite point of the orbit, when the moon is in op- 
position at B, the force with which the sun draws the moon is less 
than that with which the sun draws the earth, so that the effect of 
the solar force is to separate the moon and earth, or to increase 
their distance ; that is, it is the same as if, conceiving the earth 
not to be acted on, the sun's force drew the moon in the direction 
from E to B. This force is negative, therefore, in respect to the 
force at A, and the effect in both cases is to draw the moon from 
the earth in a direction perpendicular to the line of the quadra- 
tures. Hence, the general result is, that by the disturbing force 
of the sun, the gravity to the earth is increased at the quadratures, 
and diminished at the syzygies. It is found by calculation that the 
average amount of this disturbing force is ^ of the moon's 
gravity to the earth.* 

* Playfair. 



132 THE MOON. 

229. With these general principles in view, we may now pro- 
ceed to investigate the figure of the moon's orbit, and the irregu- 
larities to which the motions of this body are subject. 

230. The figure of the moon's orbit is an ellipse, having the earth 
in one of the foci. 

The elliptical figure of the moon's orbit, is revealed to us by ob- 
servations on her changes in apparent diameter, and in her hori- 
zontal parallax. First, we may measure from day to day the ap- 
parent diameter of the moon. Its variations being inversely as 
the distances, (Art. 163,) they give us at once the relative distance 
of each point of observation from the focus. Secondly, the va- 
riations on the moon's horizontal parallax, which also are inversely 
as the distances, (Art. 82,) lead to the same results. Observations 
on the angular velocities, combined with the changes in the lengths 
of the radius vector, afford the means of laying down a plot of the 
lunar orbit, as in the case of the sun, represented ' in figure 32. 
The orbit is shown to be nearly an ellipse, because it is found to 
have the properties of an ellipse. 

The moon's greatest and least apparent diameters are respectively 
33'.518 and 29'.365, while her corresponding changes of parallax 
are 61 '.4 and 53'.8. The two ratios ought to be equal, and we 
shall find such to be the fact very nearly, as expressed by the fore- 
going numbers ; for, 

61.4 : 53.8 : : 33.518 : 29.369. 

The greatest and least distances of the moon from the earth, 
derived from the parallaxes, are 63.8419 and 55.9164, or nearly 
64 and 56. the radius of the earth being taken for unity. Hence, 
taking the arithmetical mean, which is 59.879, we find that the 
mean distance of the moon from the ea^th is very nearly 60 times 
the radius of the earth. The point in the moon's orbit nearest 
the earth, is called her perigee ; the point furthest from the earth, 
her apogee. 

The greatest and least apparent diameters of the sun are re- 
spectively 32.583, and 31.517, which numbers express also the ratio 
of the greatest and least distances of the earth from the sun. By 
comparing this ratio with that of the distances of the moon, it will 
be seen that the latter vary much more than the former, and con- 



LUNAR IRREGULARITIES. 133 

sequently that the lunar orbit is much more eccentric than the so* 
lar. The eccentricity of the moon's orbit is in fact 0.0548, (the 
semi-major axis being as usual taken for unity) =^5- of its mean 
distance from the earth, while that of the earth is only .01685=^V 
of its mean distance from the sun. 

231. The moon 1 s nodes constantly shift their positions in the eclip- 
tic from east to west, at the rate of 19 35' per annum, returning to 
the same points in 18. 6 years. 

Suppose the great circle of the ecliptic marked out on the face 
of the sky in a distinct line, and let us observe, at any given time, 
the exact point where the moon crosses this line, which we will 
suppose to be close to a certain star ; then, on its next return to 
that part of the heavens, we shall find that it crosses the ecliptic 
sensibly to the westward of that star, and so on, further and fur- 
ther to the westward every time it crosses the ecliptic at either 
node. This fact is expressed by saying that the nodes retrograde 
on the ecliptic, and that the line which joins them, or the line of 
the nodes, revolves from east to west. 

232. This shifting of the moon's nodes implies that the lunar 
orbit is not a curve returning into itself, but that it more resem- 
bles a spiral like the curve represented in figure 49, where abc 
represents the ecliptic, and ABC the Fig. 49. 

lunar orbit, having its nodes at C and 

E, instead of A and a ; consequently, 

the nodes shift backwards through 

the arcs C and AE. The manner 

in which this effect is produced may 

be thus explained. That part of the 

solar force which is parallel to the line joining the centers of the 

sun and earth, (See Fig. 48,) is not in the plane of the moon's 

orbit, (since this is inclined to the ecliptic about 5,) except when 

the sun itself is in that plane, or when the line of the nodes being 

produced, passes through the sun. In all other cases it is oblique 

to the plane of the orbit, and may be resolved into two forces, 

one of which is at right angles to that plane, and is directed to- 

wards the ecliptic. This force of course draws the moon continu- 




134 



THE MOON. 



ally towards the ecliptic, or produces a continual deflection of the 
moon from the plane of her own orbit towards that of the earth. 
Hence the moon meets the plane of the ecliptic sooner than it 
would have done if that force had not acted. At every half revo- 
lution, therefore, the point in which the moon meets the ecliptic, 
shifts in a direction contrary to that of the moon's motion, or con- 
trary to the order of the signs. If the earth and sun were at rest, 
the effect of the deflecting force just described, would be to pro- 
duce a retrograde motion of the line of the nodes till that line was 
brought to pass through the sun, and of consequence, the plane of 
the moon's orbit to do the same, after which they would both re- 
main in their position, there being no longer any force tending to 
produce change in either. But the motion of the earth carries the 
line of the nodes out of this position, and produces, by that means, 
its continual retrogradation. The same force produces a small 
variation in the inclination of the moon's orbit, giving it an alter- 
nate increase and decrease within very narrow limits.* These 
points will be easily understood by the aid of a diagram. There- 
fore, let MN (Fig. 50,) be the ecliptic, ANB the orbit of the moon, 
the moon being in L, and N its descending node. Let the disturb- 
ing force of the sun which tends to bring it down to the ecliptic 

Fig. 50. 




be represented by L6, and its velocity in its orbit by La. Under 
the action of these two forces, the moon will describe the diago- 
nal Lc of the parallelogram ba, and its orbit will be changed from 
AN to LN' ; the node N passes to N' ; and the exterior angle at N' 
of the triangle LNN' being greater than the interior and opposite 

* Playfair. 



LUNAR IRREGULARITIES. 135 

angle at N, the inclination of the orbit is increased at the node. 
After the moon has passed the ecliptic to the south side to Z, the 
disturbing force Id produces a new change of the orbit N'/e to 
N"lf, and the inclination is diminished as at N". In general, 
while the moon is receding from one of the nodes, its inclination is 
diminishing : while it is approaching a node, the inclination is in- 
creasing.* 

233. The period occupied by the sun in passing from one of 
the moon's nodes until it comes round to the same node again, is 
called the si/nodical revolution of the node. This period is shorter 
than the sidereal year, being only about 346? days. For since 
the node shifts its place to the westward 19 35' per annum, the 
sun, in his annual revolution, comes to it so much before he com- 
pletes his entire circuit ; and since the sun moves about a degree 
a day, the synodical revolution of the node is 365 19=346, or 
more exactly, 346.619851. The time from one new moon, or 
from one full moon, to another, is 29.5305887 days. Now 19 
synodical revolutions of the nodes contain very nearly 223 of 
these periods. 

For 346.619851x19=6585.78, 

And 29.5305887X223=6585.32. 

Hence, if the sun and moon were to leave the moon's node toge- 
ther, after the sun had been round to the same node 19 times, the 
moon would have performed very nearly 223 synodical revolu- 
tions, and would, therefore, at the end of this period meet at the 
same node, to repeat the same circuit. And since eclipses of the 
sun and moon depend upon the relative position of the sun, the 
moon, and node, these phenomena are repeated in nearly the same 
order, in each of those periods. Hence, this period, consisting of 
about 18 years and 10 days, under the name of the Saros, was 
used by the Chaldeans and other ancient nations in predicting 
eclipses. 

234. The Metonic Cycle is not the same with the Saros, but 
consists of 19 tropical years. During this period the moon makes 

* Francceur, Uranog. p. 158. Robison's Phys. Astronomy, Art. 264. 



136 THE MOON. 

very nearly 235 synodical revolutions, and hence the new and full 
moons, if reckoned by periods of 19 years, recur at the same 
dates. If, for example, a new moon fell on the fiftieth day of one 
cycle, it would also fall on the fiftieth day of each succeeding cycle ; 
and, since the regulation of games, feasts, and fasts, has been 
made very extensively according to new or full moons, hence this 
lunar cycle has been much used both in ancient and modern 
times. The Athenians adopted it 433 years before the Christian 
era, for the regulation of their calendar, and had it inscribed in 
letters of gold on the walls of the temple of Minerva. Hence the 
term Golden Number, which denotes the year of the lunar cycle. 

235. The line of the apsides of the moon's orbit revolves from 
west to east through her whole orbit in about nine years. 

If, in any revolution of the moon, we should accurately mark 
the place in the heavens where the moon comes to its perigee, 
(Art. 230,) we should find, that at the next revolution, it would 
come to its perigee at a point a little further eastward than before, 
and so on at every revolution, until, after 9 years, it would come 
to its perigee at nearly the same point as at first. This fact is 
expressed by saying that the perigee, and of course the apogee, 
revolves, and that the line which joins these two points, or the line 
of the apsides, also revolves. 

The place of the perigee may be found by observing when the 
moon has the greatest apparent diameter. But as the magnitude 
of the moon varies slowly at this point, a better method of ascer- 
taining the position of the apsides, is to take two points in the or- 
bit where the Variations in apparent diameter are most rapid, and 
to find where they are equal on opposite sides of the orbit. The 
middle point between the two will give the place of the perigee. 

The angular distance of the moon from her perigee in any part 
of her revolution, is called the Mooris Anomaly. 

236. The change of place in the apsides of the moon's orbit, 
like the shifting of the nodes, is caused by the disturbing influence 
of the sun. If when the moon sets out from its perigee, it were* 
urged by no other force than that of projection, combined with its 
gravitation towards the earth, it would describe a symmetrical 



LUNAR IRREGULARITIES. 137 

curve (Art. 186,) coming to its apogee at the distance of 180. 
But as the mean disturbing force in the direction of the radius 
vector tends, on the whole, to diminish the gravitation of the 
moon to the earth, the portion of the path described in an instant 
will be less deflected from her tangent, or less curved, than if this 
force did not exist. Hence the path of the moon will not inter- 
sect the radius vector at right angles, that is, she will not arrive at 
her apogee until after passing more than 180 from her perigee, 
by which means these points will constantly shift their positions 
from west to east.* The motion of the apsides is found to be 3 
1' 20" for every sidereal revolution of the moon. 

237. On account of the greater eccentricity of the moon's orbit 
above that of the sun, the Equation of the Center., or that correc- 
tion which is applied to the moon's mean anomaly to find her true 
anomaly (Art. 200,) is much greater than that of the sun, being 
when greatest more than six degrees, (6 17' 12".7,) while that of 
the sun is less than two degrees, (1 55' 26".8.) 

The irregularities in the motions of the moon may be compared 
to those of the magnetic needle. As a first approximation, we say 
that the needle places itself in a north and south line. On closer 
examination, however, we find that, at different places, it deviates 
more or less from this line, and we introduce the first great cor- 
rection under the name of the declination of the needle. But ob- 
servation shows us that the declination alternately increases and 
diminishes every day, and therefore we apply to the declination 
itself a second correction for the diurnal variation. Finally, we 
ascertain, from long continued observations, that the diurnal va- 
riation is affected by the change of seasons, being greater in sum- 
mer than in winter, and hence we apply to the diurnal variation a 
third correction for the annual variation. 

In like manner, we shall find the greater inequalities of the 
moon's motions are themselves subject to subordinate inequalities, 
which give rise to smaller equations, and these to smaller still, to 
the last degree of refinement. 

238. Next to the equation of the center, the greatest correction 

18 * 



138 THE MOON. 

to be applied to the moon's longitude, is that which belongs to the 
Evection. The evection is a change of form in the lunar orbit, by 
which its eccentricity is sometimes increased, and sometimes 
diminished. It depends on the position of the line of the apsides 
with respect to the line of the syzygies. 

This irregularity, and its connexion with the place of the peri- 
gee with respect to the place of conjunction or opposition, was 
known as a fact to the ancient astronomers, Hipparchus and 
Ptolemy ; but its cause was first explained by Newton in con- 
formity with the law of universal gravitation. It was found, by 
observation, that the equation of the center itself was subject to a 
periodical variation, being greater than its mean, and greatest of 
all when the conjunction or opposition takes place at the perigee 
or apogee, and least of all when the conjunction or opposition 
takes place at a point half way between the perigee and apogee ; 
or, in the more common language of astronomers, the equation of 
the center is increased when the line of the apsides is in syzygy, 
and diminished when that line is in quadrature. If, for example, 
when the line of the apsides is in syzygy, we compute the moon's 
place by deducting the equation of the center from the mean 
anomaly (see Art. 200,) seven days after conjunction, the compu- 
ted longitude will be greater than that determined by actual obser- 
vation, by about 80 minutes ; but if the same estimate is made 
when the line of the apsides is in quadrature, the computed longi- 
tude will be less than the observed, by the same quantity. These 
results plainly show a connexion between the velocity of the 
moon's motions and the position of the line of the apsides with 
respect to the line of the syzygies. 

239. Now any cause which, at the perigee, should have the 
effect to increase the moon's gravitation towards the earth beyond 
its mean, and, at the apogee, to diminish the moon's gravitation 
towards the earth, would augment the difference between the 
gravitation at the perigee arid at the apogee, and consequently in- 
crease the eccentricity of the orbit. Again, any cause which at 
the perigee should have the effect to lessen the moon's gravitation 
towards the earth, and, at the apogee, to increase it, would lessen 
the difference between the two, and consequently diminish the 



LUNAR IRREGULARITIES. 139 

eccentricity of the orbit, or bring it nearer to a circle. Let us 
see if the disturbing force of the sun produces these effects. The 
sun's disturbing force, as we have sen in article 228, admits of 
two resolutions, one in the direction of the radius vector, (OM, 
Fig. 48,) the other (ON) in the direction of a tangent to the orbit. 
First, let AB be the line of the apsides in syzygy, A being the place 
of the perigee. The sun's disturbing force OM is greatest in the 
direction of the line of the syzygies ; yet depending as it does on the 
unequal action of the sun upon the earth and the moon, and being 
greater as their distance from each other is greater, it is at a mini- 
mum when acting at the perigee, and at a maximum when acting at 
the apogee. Hence its effect is to draw away the moon from the 
earth less than usual at the perigee, and of course to make her 
gravitation towards the earth greater than usual, while at the 
apogee its effect is to diminish the tendency of the moon to the 
earth more than usual; and thus to increase the disproportion be- 
tween the two distances of the moon from the focus at these two 
points, and of course to increase the eccentricity of the orbit. 
The moon, therefore, if moving towards the perigee, is brought 
to the line of the apsides in a point between its mean place and 
the earth ; or if moving towards the apogee, she reaches the line 
of the apsides in a point more remote from the earth than its mean 
place. 

Secondly, let CD be the line of the apsides, in quadrature, C 
being the place of the perigee. The effect of the sun's disturb- 
ing force is to increase the tendency of the moon towards the 
earth when in quadrature. If, however, the moon is then at her 
perigee, such increase will be less than usual, and if at her apogee, 
it will be more than usual ; hence its effect will be to lessen the 
disproportion between the two distances of the moon from the 
focus at these two points ; and of course to diminish the eccen- 
tricity of the orbit. The moon, therefore, if moving towards 
the perigee, meets the line of the apsides in a point more remote 
from the earth than the mean place of the perigee ; and if moving 
towards the apogee, in a point between the earth and the mean place 
of the apogee.* 

Woodhouse's Ast^p. 680. 



140 THE MOON. 

240. A third inequality in the lunar motions, is the Variation. 
By comparing the moon's place as computed from her mean mo- 
tion corrected for the eqifation of the center and for evection, 
with her place as determined by observation, Tycho Brahe dis- 
covered that the computed and observed places did not always 
agree. They agreed only in the syzygies lind quadratures, and 
disagreed most at a point half way between these, that is, at the 
octants. Here, at the maximum, it amounted to more than half 
a degree (35' 41. "6.) It appeared evident from examining the 
daily observations while the moon is performing her revolution 
around the earth, that this inequality is connected with the angular 
distance of the moon from the sun, and in every part of the orbit 
could be correctly expressed by multiplying the maximum value 
as given above, into the sine of twice the angular distance between 
the sun and the moon. It is, therefore, at the conjunctions and 
quadratures, and greatest at the octants. Tycho Brahe knew the 
fact : Newton investigated the cause. 

It appears by article 228, that the sun's disturbing force can be 
resolved into two parts, one in the direction of the radius vector, 
the other at right angles to it in the direction of a tangent to the 
moon's orbit. The former, as already explained, produces the 
Evection: the latter produces the Variation. This latter force 
will accelerate the moon's velocity, in every point of the quadrant 
which the moon describes in moving from quadrature to conjunc- 
tion, or from C to A, (Fig. 48,) but at an unequal rate, the 
acceleration being greatest at the octant, and nothing at the quad- 
rature and the conjunction ; and when the moon is past conjunction, 
the tangential force will change its direction and retard the moon's 
motion. All these points will be understood by inspection of 
figure 48. 

241. A fourth lunar inequality is the Annual Equation. This 
depends on the distance of the earth (and of course the moon) 
from the sun. Since the disturbing influence of the sun has a 
greater effect in proportion as the sun is nearer,* consequently all 
the inequalities depending on this influence must vary at diiTerent 

* Varying reciprocally as the cube of the sun's distance from the earth. 



LUNAR IRREGULARITIES. 141 

seasons of the year. Hence, the amount of this effect due to any 
particular time of the year is called the Annual Equation. 

242. The foregoing are the largest of the inequalities of the 
n\ oon's motions, and may serve as specimens of the corrections that 
are to be applied to the mean place of the moon in order to find 
her true place. These were first discovered by actual observa- 
tion ; but a far greater number, though most of them are exceed- 
ingly minute, have been made known by the investigations of Phys- 
ical Astronomy, in following out all the consequences of universal 
gravitation. In the best tables, about 30 equations are applied to 
the mean motions of the moon. That is, we first compute the 
place of the moon on the supposition that she moves uniformly 
in a circle. This gives us her mean place. We then proceed, 
by the aid of the Lunar Tables, to apply the different corrections, 
such as the equation of the center, evection, variation, the annual 
equation, and so on, to the number of 28. Numerous as these 
corrections appear, yet La Place informs us, that the whole num- 
ber belonging to the moon's longitude is no less than 60 ; and 
that to give the tables all the requisite degree of precision, addi- 
tional investigations will be necessary, as extensive at least as 
those already made.* The best tables in use in the time of Tycho 
Brahe, gave the moon's place only by a distant approximation. 
The tables in use in the time of Newton, (Halley's tables,) approxi- 
mated within 7 minutes. Tables at present in use give the moon's 
place to 5 seconds. These additional degrees of accuracy have 
been attained only by immense labor, and by the united efforts of 
Physical Astronomy and the most refined observations. 

243. The inequalities of the moon's motions are divided into 
periodical and secular. Periodical inequalities are those which 
are completed in comparatively short periods, like evection and 
variation: Secular inequalities are those which are completed 
only in very long periods, such as centuries or ages. Hence the 
corresponding terms periodical equations, and secular equations. 
As an example of a secular inequality, we may mention the ac* 

* Syst. du Monde, 1. iv, c. 5. 



142 THE MOON. 

celeration of the moorfs mean motion. It is discovered, that the 
moon actually revolves around the earth in less time now than 
she did in ancient times. The difference however is exceedingly 
small, being only about 10" m a century, but increases from century 
to century as the square of the number of centuries from a given 
epoch. This remarkable fact was discovered by Dr. Halley.* In a 
lunar eclipse the moon's longitude differs from that of the sun, at the 
middle of the eclipse, by exactly 180 ; and since the sun's lon- 
gitude at any given time of the year is known, if we can learn 
the day and hour when an eclipse occurs, we shall of course know 
the longitude of the sun and moon. Now in the year 721 before 
the Christian era, on a specified day and hour, Ptolemy records a 
lunar eclipse to have happened, and to have been observed by 
the Chaldeans. The moon's longitude, therefore, for that time is 
known ; and as we know the mean motions of the moon at pre- 
sent, starting from that epoch, and computing, as may easily be 
done, the place which the moon ought to occupy at present at any 
given time, she is found to be actually nearly a degree and a half 
in advance of that place. Moreover, the same conclusion is 
derived from a comparison of the Chaldean observations with those 
made by an Arabian astronomer of the tenth century. 

This phenomenon at first led astronomers to apprehend that the 
moon encountered a resisting medium, which, by destroying at 
every revolution a small portion of her projectile force, would 
have the effect to bring her nearer and nearer to the earth and 
thus to augment her velocity. But in 1786, La Place demon- 
strated that this acceleration is one of the legitimate effects of the 
sun's disturbing force, and is so connected with changes in the 
eccentricity of the earth's orbit, that the moon will continue to be 
accelerated while that eccentricity diminishes, But when the eccen- 
tricity has reached its minimum (as it will do after many ages) 
and begins to increase, then the moon's* motion will begin to be 
retarded, and thus her mean motions will oscillate forever about a 
mean value. * 

244. The lunar inequalities which have been considered are such 



* Astronomer Royal of Great Britain, and co temporary with Sir Isaac Newton. 



ECLIPSES. 143 

only as affect the moon's longitude ; but the sun's disturbing force 
also causes inequalities in the moon's latitude and parallax. Those 
of latitude alone require no less than twelve equations. Since 
the moon revolves in an orbit inclined to the ecliptic, it is easy to 
see that the oblique action of the sun must admit of a resolution 
into two forces, one of which being perpendicular to the moon's 
orbit, must effect changes in her latitude. Since also several of the 
inequalities already noticed involve changes in the length of the 
radius vector, it is obvious that the moon's parallax must be sub- 
ject to corresponding perturbations. 



CHAPTER VII. 

ECLIPSES. 

245. AN eclipse of the moon happens, when the moon in its 
revolution about the earth, falls into the earth's shadow. An 
eclipse of the sun happens, when the moon, coming between the 
earth and the sun, covers either a part or the whole of the solar 
disk. An eclipse of the sun can occur only at the time of con- 
junction, or new moon ; and an eclipse of the moon, only at the 
time of opposition, or full moon. Were the moon's orbit in the 
same plane with that of the earth, or did it coincide with the 
ecliptic, then an eclipse of the sun would take place at every 
conjunction, and an eclipse of the moon at every opposition ; for 
as the sun and earth both lie in the ecliptic, the shadow of the 
earth must also extend in the same plane, being of course always 
directly opposite to the sun ; and since, as we shall soon see, the 
length of this shadow is much greater than the distance of the 
moon from the earth, the moon, if it revolved in the plane of the 
ecliptic, must pass through the shadow at every full moon. For 
similar reasons, the moon would occasion an eclipse of the sun, 
partial or total, in some portions of the earth at every new moon. 
But the lunar orbit is inclined to the ecliptic about 5, so that the 
center of the moon, when she is furthest from her node, is 5 from 



144 THE MOON. 

the axis of the earth's shadow (which is always in the ecliptic ;) 
and, as we shall show presently, the greatest distance to which the 
shadow extends on each side of the ecliptic, that is, the greatest 
semi-diameter of the shadow, where the moon passes through it, 
is only about of a degree, while the semi-diameter of the moon's 
disk is only about | of a degree ; hence the two semi-diame- 
ters, namely, that of the moon and the earth's shadow, cannot 
overlap one another, unless, at the time of new or full moon, the 
sun is at or very near the moon's node. In the course of the sun's 
apparent revolution around the earth once a year, he is succes- 
sively in every part of the ecliptic ; consequently, the conjunctions 
and oppositions of the sun and moon may occur at any part of the 
ecliptic, either when the sun is at the moon's node, (or when he 
is passing that point of the celestial vault on which the moon's 
node is projected as seen from the earth ;) or they may occur 
when the sun is 90 from the moon's node, where the lunar and 
solar orbits are at the greatest distance from each other; or, finally, 
they may occur at any intermediate point. Now the sun, in his 
annual revolution, passes each of the moon's nodes on opposite 
sides of the ecliptic, and of course at opposite seasons of the 
year ; so that, for any given year, the eclipses commonly happen 
in two opposite months, as January and July, February and 
August, May and November. These, therefore, are called Node 
Months. 

246. If the sun were of the same size with the earth, the shadow 
of the earth would be cylindrical and infinite in length, since the 
tangents drawn from the sun to the earth (which form the bounda- 
ries of the shadow) would be parallel to each other ; but as the 
sun is a vastly larger body than the earth, the tangents converge 
and meet in a point at some distance behind the earth, forming a 
cone of which the earth is the base, and whose vertex (and of 
course its axis) lies in the ecliptic. A little reflection will also 
show us, that the form and dimensions of the shadow must be 
affected by several circumstances ; that the shadow must be of 
the greatest length and breadth when the sun is furthest from the 
earth ; that its figure will be slightly modified by the spheroidal 
figure of the earth ; and that the moon, being, at the time of it? 



ECLIPSES. 145 

opposition, sometimes nearer to the earth, and sometimes further 
from it, will accordingly traverse it at points where its breadth 
varies more or less. 

247. The semi-angle of the cone of the earth's shadow, is equal 
to the sun's apparent semi-diameter, minus his horizontal par- 
allax. 

Let AS (Fig. 51,) be the semi-diameter of the sun, BE that of 
the earth, and EC the axis of the earth's shadow. Then the 
semi-angle of the cone of the earth's shadow ECB=AES-EAB, 

Fig. 51. 
A 




of which AES is the sun's semi-diameter and EAB his horizontal 
parallax ; and as both these quantities are known, hence the angle 
at the vertex of the shadow becomes known. Putting 6 for the 
the sun's semi-diameter, andjp for his horizontal parallax, we have 
the semi-angle of the earth's shadow ECB= p. 

248. At the mean distance of the earth from the sun, the length 
of the earth's shadow is about 860,000 miles, or more than three times 
the distance of the moon from the earth. 

In the right angled triangle ECB, right angled at B, the angle 
ECB being known, and the side EB, we can find the side EC. 

For sin. (<$-) : EB : : R : EC= -^? . This value will vary 

sin. ( p) 

with the sun's semi-diameter, being greater as that is less. Its 
mean value being 16' 1".5 and the sun's horizontal parallax being 
8".6, 5-^=15' 52".9, and EB=3956.2. Hence, 

Sin. 15' 53" : Rad. : : 3956.2 : 856,275. 

Since the distance of the moon from the earth is 238,545 miles, 
the shadow extends about 3.6 times as far as the moon, and con- 

19 



146 THE MOON. 

sequently, the moon passes the shadow towards its broadest part, 
where its breadth is much more than sufficient to cover the moon's 
disk. 

249. The average breadth of the eartKs shadow where it eclipses 
the moon is almost three times the moon's diameter. 

Let mm' (Fig. 51,) represent a section of the earth's shadow 
where the moon passes through it, M being the center of the cir- 
cular section. Then the angle MEm will be the angular breadth 
of half the shadow. But, 

MEm=BmE BCE ; that is, since BmE is the moon's horizon- 
tal parallax, (Art. 82,) and BCE equals the sun's semi-diameter 
minus his horizontal parallax (<5 p,) therefore, putting P for the 
moon's horizontal parallax, we have 

MEwi = P-(S-/>)=P+.P *; that is, since P=57' 1" and 
<5_p:=:15' 52". 9, therefore, 57' I" 15' 52".9=41' 8".l, which is 
nearly three times 15' 33", the semi-diameter of the moon. Thus, 
it is seen how, by the aid of geometry, we learn to estimate vari- 
ous particulars respecting the earth's shadow, by means of simple 
data derived from observation. 

250. The distance of the moon from her node when she just 
touches the shadow of the earth, in a lunar eclipse, is called the 
Lunar Ecliptic Limit ; and her distance from the node in a solar 
eclipse, when the moon just touches the solar disk, is called the 
Solar Ecliptic Limit. The Limits are respectively the furthest 
possible distances from the node at which eclipses can take place. 

251. The Lunar Ecliptic Limit is nearly 12 degrees. 

Let ON (Fig. 52,) be the sun's path, MN the moon's, and N the 
node. Let Ca be the semi-diameter of the earth's shadow, and 
Ma the semi-diameter of the moon. Since Cand Ma are known 

Fig. 52. 




ECLIPSES. 147 

quantities, their sum CM is also known. The angle at N is 
known, being the inclination of the lunar orbit to the ecliptic. 
Hence, in the spherical triangle MCN, right angled at M,* by 
Napier's theorem, (Art. 132, Note,) 

Rad.xsin. CM^sin. CNxsin. MNC. 

The greatest apparent semi-diameter of the earth's shadow 
where the moon crosses it, computed by article 249, is 45' 52", 
and the moon's greatest apparent semi-diameter, is 16' 45". 5, 
which together, give MC equal to 62' 37". 5. Taking the incli- 
nation of the moon's orbit, or the angle MNC (what it generally 
is in these circumstances) at 5 17', and we have Rad.xsin. 

62' 37".5=sin. CNxsin. 5 17', or sin. C N= Rad ' * sin ' 62/ 37// - 5 

sin. 5 17' 

and CN=1 1 25' 40".f This is the greatest distance of the moon from 
her node at which an eclipse of the moon can take place. By 
varying the value of CM, corresponding to variations in the dis- 
tances of the sun and moon from the earth, it is found that if NC 
is less than 9, there must be an eclipse ; but between this and the 
limit, the case is doubtful. 

When the moon's disk only comes in contact with the earth's 
shadow, as in figure 52 , the phenomenon is called an appulse, 
when only a part of the disk enters the shadow, the eclipse is 
said to be partial, and total if the whole of the disk enters the 
the shadow. The eclipse is called central when the moon's center 
coincides with the axis of the shadow, which happens when the 
moon at the time of the eclipse is exactly at her node. 

252. Before the moon enters the earth's shadow, the earth be- 
gins to intercept from it portions of the sun's light, gradually in- 
creasing until the moon reaches the shadow. This partial light is 
called the moon's Penumbra. Its limits are ascertained by drawing 
the tangents AC'B' and A'C'B. (Fig. 51 .) Throughout the space 
included between these tangents more or less of the sun's light is 
intercepted from the moon by the interposition of the earth ; for 

* The line CM is to be regarded as the projection of the line which connects the 
centers of the moon and section of the earth's shadow, as seen from the earth, 
t Woodhouse's Astronomy, p. 718. 



148 THE MOON. 

it is evident, that as the moon moves towards the shadow, she 
would gradually lose the view of the sun, until, on entering the 
shadow, the sun would be entirely hidden from her. 

253. The semi-angle of the Penumbra equals the sun's semi- 
diameter and horizontal parallax, or 8+p. 

The angle hC'M (Fig. 51,)=AC'S=AES+B'AE. But AES is 
the sun's semi-diameter, and B'AE is the sun's horizontal parallax, 
both of which quantities are known. 

254. The semi-angle of a section of the Penumbra, where the 
moon crosses it, equals the moon's horizontal parallax, plus the sun's, 
plus the sun's semi-diameter. 

The angle AEM (Fig. 51,) =EhC'+EC'h. But EhC'=P, the 
moon's horizontal parallax, and EC'h =8+p (Art. 253,) /. hEM 
=P+p+<5, all which are likewise known quantities. 

Hence, by means of these few elements, which are known from 
observation, we ascend to .a complete knowledge of all the par- 
ticulars necessary to be kijiown respecting the moon's penumbra. 

255. In the preceding investigations, we have supposed that 
the cone of the earth's shadow is formed by lines drawn from the 
sun, and touching the earth's surface. But the apparent diameter 
of the shadow is found by observation to be somewhat greater than 
would result from this hypothesis. The fact is accounted for by 
supposing that a portion of the solar rays which graze the earth's 
surface are absorbed and extinguished by the lower strata of the 
atmosphere. This amounts to the same thing as though the earth 
were larger than it is, in which case the moon's horizontal parallax 
would be increased ; and accordingly, in order that theory and 
observation may coincide, it is found necessary to increase the 
parallax by F V- 

256. In a total eclipse of the moon, its disk is still visible, 
shining with a dull red light. This light cannot be derived di- 
rectly from the sun, since the view of the sun is completely hid- 
den from the moon ; nor by reflexion from the earth, since the 
illuminated side of the earth is wholly turned from the moon ; but 



ECLIPSES. 149 

it is owing to refraction by the earth's atmosphere, by which a few 
scattered rays of the sun are bent round into the earth's shadow 
and conveyed to the moon, sufficient in number to afford the feeble 
light in question. 

257. In calculating an eclipse of the moon, we first learn from 
the tables in what month the sun, at the time of full moon in that 
month, is near the moon's node, or within the lunar ecliptic limit. 
This it must evidently be easy to determine, since the tables ena- 
ble us to find both the longitudes of the nodes, and the longitudes 
of the sun and moon, for every day of the year. Consequently, 
we can find when the- sun has nearly the same longitude as one of 
the nodes, and also the precise moment when the longitude of the 
moon is 180 from that of the sun, for this is the time of opposition, 
from which may be derived the time of the middle of the eclipse. 
Having the time of the middle of the eclipse, and the breadth 
of the shadow, (Art. 249,) and knowing, from the tables, the rate 
at which the moon moves per hour faster than the shadow, we can 
find how long it will take her to traverse half the breadth of the 
shadow ; and this time subtracted from the time of the middle 
of the eclipse, will give the beginning, and added to the time of 
the middle will give the end of the eclipse. Or if instead of the 
breadth of the shadow, we employ the breadth of the penumbra 
(Art. 253,) we may find, in the same manner, when the moon 
enters and when she leaves the penumbra. We see, therefore, 
how by having a few things known by observation, such as the 
sun and moon's semi-diameters, and their horizontal parallaxes, 
we rise, by the aid of trigonometry, to the knowledge of various 
particulars respecting the length and breadth of the shadow and 
of the penumbra. These being known, we next have recourse to 
the tables which contain all the necessary particulars respecting 
the motions of the sun and moon, together with equations or cor- 
rections, to be applied for all their irregularities. Hence it is com- 
paratively an easy task to calculate with great accuracy an eclipse 
of the moon. 

/ 

258. Let us then see how we may find the exact time of the be- 
ginning, end, duration, and magnitude, of a lunar eclipse. 



150 THE MOON. 

Let NG (Fig 53,) be the ecliptic, and Nog- the moon's orbit, the 
sun being in A* when the moon is in opposition at a ; let N be 
the ascending node, and Aa the moon's latitude at the instant 

Fig. 53. 




of opposition. An hour afterwards the sun will have passed to 
A', and the moon to g, when the difference of longitude of the two 
bodies will be GA'. Then gli is the moon's hourly motion in lati- 
tude, and ah ' her hourly motion in longitude. As the character 
and form of the eclipse will depend solely upon the distances 
between the centers of the sun and moon, that is, upon the line 
g-A', instead of considering the two bodies as both in motion, 
we may suppose the sun at rest in A, and the moon as advancing 
with a motion equal to the difference between its rate and that 
of the sun, a supposition which will simplify the calculation. 
Therefore, draw gd parallel and equal to A'A, join dA, and this 
line being equal to gA', the two bodies will be in the same relative 
situation as if the sun were at A' and the moon atg*. Join da and 
produce the line da both ways, cutting the ecliptic in F ; then 
6?F will be the moon's Relative Orbit. Hence ai=ah AA'=the 
difference of the hourly motions of the sun and moon, that is, the 
moon's relative motion in longitude, and di the moon's hourly 
motion in latitude. 

Draw CD (Fig. 54,) to represent the ecliptic, and let A be the 
place of the sun. As the tables give the computation of the 
moon's latitude at every instant, consequently, we may take from 
them the latitude corresponding to the instant of opposition, and 
to one hour later ; and we may take also the sun's and moon's 
hourly motions in longitude. Take AD, AB, each equal to fhe 
relative motion, and Aa=the latitude in opposition, Dd= the lati- 



* It will be remarked that the point A really represents the center of the earth's 
shadow ; but as the real motions of the shadow are the same with the assumed motion* 
of the sun, the latter are used in conformity with the language of the tables. 



ECLIPSES. 

Fig. 54. 



151 




D F 



F' 



A M 



B C 



tude one hour afterwards ; join da and produce the line da both 
ways, and it will represent the moon's' relative orbit. Draw B6 
at right angles to CD, and it will be the latitude an hour before 
opposition. At the time of the eclipse, the apparent distance of 
the center of the shadow from the moon is very small ; conse- 
quently, CD, cd, Dd, &c. may be regarded as straight lines. 
During the short interval between the beginning and end of an 
eclipse, the motion of the sun, and consequently that of the cen- 
ter of the shadow, may likewise be regarded as uniform. 

259. The various particulars that enter into the calculation of 
an eclipse are called its Elements ; and as our object is here merely 
to explain the method of calculating an eclipse of the moon, (refer- 
ring to the Supplement for the actual computation,) we may take 
the elements at their mean value. Thus, we will consider cd as 
inclined to CD 5 9', the moon's horizontal parallax as 58', its semi- 
diameter as 16', and that of the earth's shadow as 42'. The line 
Am perpendicular to cd gives the point m for the place of the 
moon at the middle of the eclipse, for this line bisects the chord, 
which represents the path of the moorr through the shadow ; and 
wM, perpendicular to CD, gives AM for the time of the middle 
of the eclipse before opposition, the number of minutes before op- 
position being the same part of an hour that AM is of AB.* From 
the center A, with a radius equal to that of the earth's shadow 
(42') describe the semi-circle BLF, and it will represent the pro- 
jection of the shadow traversed by the moon. With a radius 
equal to the semi-diameter of the shadow and that of the moon 



* The situation of the moon when at m is called orbit opposition ; and her situation 
when at a, ecliptic opposition. 



152 THE MOON. 

(=42'+16'=58') and with the center A, mark the two points c and 
f on the relative orbit, and they will be the places of the center 
of the moon at the beginning and end of the eclipse. The per- 
p'endiculars cC,/F, give the times AC and AF of the commence- 
ment and the end of the eclipse, and CM, or MF gives half the 
duration. From the centers c and f with a radius equal to the 
semi-diameter of the moon (16') describe circles, and they will 
each touch the shadow, (Euc. 3.12.) indicating the position of the 
moon at the beginning and end of the eclipse. If the same circle 
described from m-is wholly within the shadow, the eclipse will be 
total; if it is only partly within the shadow, the eclipse will be 
partial. With the center A, and radius equal to the semi-diame- 
ter of the shadow minus that of the moon (42' 16'=26') mark 
the two points c',/\ which will give the places of the center of the 
moon, at the beginning and end of total darkness, and MC', MF' 
will give the corresponding times before and after the middle of 
the eclipse. Their sum will be the duration of total darkness. 

260. If the foregoing projection be accurately made from a scale, 
the required particulars of the eclipse may be ascertained by 
measuring on the same scale, the linds which respectively repre- 
sent them ; and we should thus obtain a near approximation to the 
elements of the eclipse. A more accurate determination of these 
elements may, however, be obtained by actual calculation. The 
general principles of the calculation will be readily understood. 

First, knowing ai, (Fig. 53,) the moon's relative longitude, and 
di, her latitude, we find the angle dai, which is the inclination of 
the moon's relative orbit. But dai a Am ; and, in the triangle 
aAm, we have the angle at A, and the side A<z, being the moon's 
latitude at the time of opposition, which is given by the tables. 
Hence we can find the side Am. In the triangle AmM, (Fig. 54,) 
having the side Am and the angle AmM (=aAm) we can find AM 
= the arc of relative longitude described by the moon from the 
time of the middle of the eclipse to the time of opposition ; and 
knowing the moon's hourly motion in longitude, we can convert 
AM into time, and this subtracted from the time of opposition 
gives us the time of the middle of the eclipse. 



ECLIPSES. 153 

Secondly, since we know the length of the line Ac* (Fig. 54) 
and can easily find the angle cAC, we can thus obtain the side 
AC ; arid AC AM=MC, which arc, converted into time by com- 
paring it with the moon's hourly motion in longitude, gives us, 
when subtracted from the time of the middle of the eclipse, the 
time of the beginning of the eclipse, or when added to that of the 
middle, the time of the end of the eclipse. The sum of the two 
equals the whole duration. 

Thirdly, by a similar method we calculate the value of MC', 
which converted into time, and subtracted from, the time of the 
middle of the eclipse, gives the commencement of total darkness, or 
when added gives the tnd of total darkness. Their sum is the 
duration of total darkness. 

Fourthly, the quantity of the eclipse is determined by supposing 
the diameter of the moon divided into twelve equal parts called 
Digits, and finding how many such parts lie within the shadow, 
at the time when the centers of the moon and the shadow are 
nearest to each other. Even when the moon lies wholly within 
the shadow, the quantity of the eclipse is still expressed by the num- 
ber of digits contained in that part of the line which joins the cen- 
ter of. the shadow and the center of the moon, which is intercepted 
between the edge of the shadow and the inner edge of the moon. 

Thus in figure 54, the number of digits eclipsed, equals 

T2 7 " 

AoAn Ao(Amnm) . . . 

= = " 1 , an expression containing only known 

rV/i/ TV" 

quantities. 

261. The foregoing will serve as an explanation of the general 
principles, on which proceeds the calculation of a lunar eclipse. 
The actual methods practiced employ many expedients to facili- 
tate the process, and to insure the greatest possible accuracy, the 
nature of which are explained and exemplified in Mason's Supple- 
ment to this work. 

262. The leading particulars respecting an JEcLiPSE OF THE 
SUN, are ascertained very nearly like those of a lunar eclipse. The 



* This line is not represented in the figure, but may be easily imagined. 

20 



154 THE MOON. 

shadow of the moon travels over a portion of the earth, as the 
shadow of a small cloud, seen from an eminence in a clear day, 
rides along over hills and plains. Let us imagine ourselves stand- 
ing on the moon ; th'en we shall see the earth partially eclipsed by 
the shadow of the moon, in the same manner as we now see the 
mogn eclipsed by the earth's shadow ; and we might proceed to 
find the length of the shadow, its breadth where it eclipses the 
earth, the breadth of the penumbra, and its duration and quantity, 
in the same way as we have ascertained these particulars for an 
eclipse of the moon. 

But, although the general characters of a solar eclipse might be 
investigated on these principles, so far as respects the earth at 
large, yet as the appearances of the same eclipse of the sun are 
very different at different places on the earth's surface, it is neces- 
sary to calculate its peculiar aspects for each place separately, a 
circumstance which makes the calculation of a solar eclipse much 
more complicated and tedious than of an eclipse of the moon. 
The moon, when she enters the shadow of the earth, is deprived 
of the light of the part immersed, and that part appears black 
alike to all places where the moon is above the horizon. But it is 
not so with a solar eclipse. We do not see this by the shadow 
cast on the earth, as we should do if we stood on the moon, but 
by the interposition of the moon between us and the sun ; and the 
sun may be hidden from one observer while he is in full view of 
another only a few miles distant. Thus, a small insulated cloud 
sailing in a clear sky, will, for a few moments, hide the sun from 
us, and from a certain space near us, while all the region around 
is illuminated. 

263. We have compared the motion of the moon's shadow over 
the surface of the earth to that of a cloud ; but its velocity is in- 
comparably greater. The mean motion of the moon around the 
ea^rth is about 33' per hour ; but 33' of the moon's orbit is 2280 
miles, and the shadow moves of course at the same rate, or 2280 
miles per hour, traversing the entire disk of the earth in less than 
four hours. This is the velocity of the shadow when it passes 
perpendicularly over the earth ; when the direction of the axis of 
the shadow is oblique to the earth's surface, the velocity is increased 



ECLIPSES. 155 

in proportion of radius to the sine of obliquity. Thus the shadows 
of evening have a far more rapid motion fchan those of noon-day. 
Let us endeavor to form a just conception of the manner in 
which these three bodies, the sun, the earth, and the moon, are 
situated with respect to each other at the time of a solar eclipse. 
First, suppose the conjunction to take place at the node. Then 
the straight line which connects the centers of the sun and the 
earth, also passes through the center of the moon, and coincides 
with the axis of its shadow ; and, since the earth is bisected by 
the plane of the ecliptic, the shadow would traverse the earth in 
the direction of the terrestrial ecliptic, from west to east, passing 
over the middle regions of the earth. Here the diurnal motion of 
the earth being in the same direction with the shadow, but with a 
less velocity, the shadow will appear to move with a speed equal 
only to the difference between the two. Secondly, suppose the 
moon is on the north side of the ecliptic at the time of conjunction, 
and moving towards her descending node, and that the conjunc- 
tion takes place just within the solar ecliptic limit, say 16 from the 
node. The shadow will now not fall in the plane of the ecliptic, 
but a little northward of it, so as just to graze the earth near the 
pole of the ecliptic. The nearer the conjunction comes to the 
node, the further the shadow will fall from the pole of the ecliptic 
towards the equatorial regions. In certain cases, the shadow 
strikes beyond the pole of the earth ; and then its easterly motion 
being opposite to the diurnal motion of the places which it traver- 
ses, consequently its velocity is greatly increased, being equal to 

the sum of both. 

. 

264. After these general considerations, we will now examine 
more particularly the method of investigating the elements of a 
solar eclipse. 4 

The length of the moon's shadow, is the first object of inquiry. 
The moon, as well as the earth, is at different distances from the 
sun at different times, and hence the length of her shadow varies, 
being always greatest wKen she is furthest from the sun. Also, 
since her distance from the earth varies, the section of the moon's 
shadow made by the earth, is greater in proportion as the moon is 



156 THE MOON. 

nearer the earth. The greatest eclipses of the sun, therefore, 
happen when the sun is in apogee,* and the moon in perigee. 

265. When the moon is at her mean distance from the earth, and 
from the sun, her shadow nearly reaches the earth's surface. 

Let S (Fig. 55,) represent the sun, D the moon, and T the 
earth. Then, the semi-angle of the cone of the moon's shadow, 
DKR, will, as in the case of the earth, (Art. 247,) equal SDR 
DRK, of which SDR is the sun's apparent semi-diameter, as seen 
from the moon, and DRK, is the sun's horizontal parallax at the 
moon. Since, on account of the great distance of the sun, corn- 
Fig. 55. 




II 

pared with that of the moon, the semi-diameter of the sun as seen 
from the moon, must evidently be very nearly the same as 
when seen from the earth, and since on account of the minute- 
ness of the moon's semi-diameter when seen from the sun, the 
sun's horizontal parallax at the moon must be very small, we might, 
without much error, take the sun's apparent semi-diameter from 
the earth, as equal to the semi-angle of the cone of the moon's 
shadow ; but, for the sake of greater accuracy, let us estimate the 
value of the sun's semi-diameter and horizontal parallax at the 
moon. 

Now, SDR : STR : : ST : SDf : : 400 : 399 ; hence SDR = 

STR=1.0025 STR ; and the sun's mean semi-diameter STR 
399 

being 16.025, hence SDR^1.0025xl6.025 = 16.065= 3 16' 3".9. 

Again, since parallax is inversely as the distance, the sun's hor- 
izontal parallax at the moon, is on account of her being nearer the 
sun T ^ greater than at the earth ; but on account of her inferior 

* The sun is said to be in apogee, when the earth is in aphelion, 
t The apparent magnitude of an object being reciprocally as its distance from the 
eye. See Note, p. 86. 



ECLIPSES. 157 

size it is 3.}$ loss than at the earth. Hence, increasing the sun's 
horizontal parallax at the earth by the former fraction, and dimin- 

400 2160 
ishing it by the latter, we have x x 9" =2". 5= the sun's 



horizontal parallax at the moon. Therefore, the semi-angle of the 
cone of the moon's shadow, which, as appears above, equals 
SDR DRK, equals 16 3".9 2".5=16' 1 '.4, which so nearly 
equals the sun's apparent semi-diameter, as seen from the earth, 
that we may adopt the latter as the value of the semi-angle of the 
shadow. Hence, sin. 16' 1".5 : 1080 (BD) : : Rad. : DK=231690. 
But the mean distance of the moon from the surface of the earth 
is 238545 3956=234589, which exceeds a little the mean length 
of the shadow as above. 

But when the moon is nearest the earth her distance from the 
center of the earth is only 221148 miles; and when the earth is 
furthest from the sun, the sun's apparent semi-diameter is only 
15' 45". 5. By employing this number in the foregoing estimate. 
we shall find the length of the shadow 235630 miles ; and 
235630 221148=14482, the distance which the moon's shadow 
may reach beyond the center of the earth. 

266. The diameter of the moon's shadow where it traverses the 
earth, is, at its maximum, about 170 miles.* 

In the triangle eTK, the angle at K=15' 45".5 (Art. 265,) the 
side Te-3956, and TK=14482. 

Or, 3956 : 14482 : : sin. 15' 45".5 : sin. 57' 41".5. 

And 57' 41".5+15' 45".5=1 13' 27"=dTe, or the arc de. 

And 2de=2 26' 54"=en. 

Hence 360 : 2.45 (=2 26' 54") : : 24899f : 170 (nearly). 

267. The greatest portion of the earth's surface ever covered by 
the moon's penumbra, is about 4393 miles. 

The semi-angle of the penumbra BID=BSD+SBR, of which 
BSD the sun's horizontal parallax at the moon =2 /7 .5, and SBR 
the sun's apparent semi-diameter =16' 3".9, and hence BID is 

* This supposes the conjunction to take place at the node, and the 'shadow to strike 
the earth perpendicularly to its surface ; where it strikes obliquely, the section may be 
greater than this. 

t The equatorial circumference. 



158 THE MOON. 

known. The moon's apparent semi-diameter BGD=16' 45' .5. 
Therefore GDT is known, as likewise DT and TG. Hence the 
angle GTd may be found, and the arc cZG and its double GH, 
which equals the angular breadth of the penumbra. It may be 
converted into miles by stating a proportion as in article 266. 
On making the calculation it will be found to be 4393 miles. 

268. The apparent diameter of the moon is sometimes larger 
than that of the sun, sometimes smaller, and sometimes exactly 
equal to it. Suppose an observer placed on the right line which 
joins the centers of the sun and moon ; if the apparent diameter of 
the moon is greater than that of the sun, the eclipse will be total. 
If the two diameters are equal, the moon's shadow just reaches the 
earth, and the sun is hidden but for a moment from the view of 
spectators situated in the line which the vertex of the shadow de- 
scribes on the surface of the earth. But if, as happens when the 
moon comes to her conjunction in that part of her orbit which is 
towards her apogee, the moon's diameter is less than the sun's, 
then the observer will see a ring of the sun encircle the moon, 
constituting an annular eclipse. (Fig. 55'.) 

Fig. 55'. 




269. Eclipses of the sun are modified by the elevation of the 
moon above the horizon, since its apparent diameter is augmented 



ECLIPSES. 159 

as its altitude is increased, (Art. 217.) This effect, combined with 
that of parallax, may so increase or diminish the apparent distance 
between the centers of the sun and moon, that from this cause 
alone, of two observers at a distance from each other, one might 
see an eclipse which was not visible to the other.* If the hori- 
zontal diameter of the moon differs but little from the apparent 
diameter of the sun, the case might occur where the eclipse would 
be annular over the places where it was observed morning and 
evening, but total where it was observed at mid-day. 

The earth in its diurnal revolution and the moon's shadow both 
move from west to east, but the shadow moves faster than the 
earth ; hence the moon overtakes the sun on its western limb and 
crosses it from west to east. The excess of the apparent diame- 
ter of the moon above that of the sun in a total eclipse is so small, 
that total darkness seldom continues longer than four minutes, and 
can never continue so long as eight minutes. An annular eclipse 
may last 12m. 24s. 

Since the sun's ecliptic limits are more than 17 and the moon's 
less than 12, eclipses of the sun are more frequent than those of 
the moon. Yet lunar eclipses being visible to every part of the 
terrestrial hemisphere opposite to the sun, while those of the sun 
are visible only to the small portion of the hemisphere on which 
the moon's shadow falls, it happens that for any particular place 
on the earth, lunar eclipses are more frequently visible than solar. 
In any year, the number of eclipses of both luminaries cannot be 
less than two nor more than seven : the most usual number is four, 
and it is very rare to have more than six. A total eclipse of the 
moon frequently happens at the next full moon after an eclipse of 
the sun. For since, in an eclipse of the sun, the sun is- at or near 
one of the moon's nodes, the earth's shadow must be at or near 
the other node, and may not have passed so far from the node as 
the lunar ecliptic limits, before the moon overtakes it. 

270. It has been observed already, that were the spectator on 
the moon instead of on the earth, he would see the earth eclipsed 
by the moon, and the calculation of the eclipse would be very sim- 
ilar to that of a luna : eclipse ; but to an observer on the earth the 

* Biot, Ast. Phys. p. 401. 






160 THE MOON. 



eclipse does not of course begin when the earth first enters the 
moon's shadow, and it is necessary to determine not only what 
portion of the earth's surface .will be covered by the moon's sha- 
dow, but likewise the path described by its center relative to va- 
rious places on the surface of the earth. This is known when the 
latitude and longitude of the center of the shadow on the earth, is 
determined for each instant. The latitude and longitude of the 
moon are found on the supposition that the spectator views it from 
the center of the earth, whereas his position on the surface changes, 
in consequence of parallax, both the latitude and longitude, and 
the amount of these changes must be accurately estimated, before 
the appearance of the eclipse at any particular place can be fully 
determined. 

The details of the method of calculating a solar eclipse cannot 
be understood in any way so well, as by actually performing the 
process according to a given example. For such details therefore 
the reader is referred to the Supplement. 

271. In total eclipses of the sun, there has sometimes been ob- 
served a remarkable radiation of light from the margin of the sun. 
This has been ascribed to an illumination of the solar atmosphere, 
but it is with more probability owing to the zodiacal light (Art. 
152.) which at that time is projected around the sun, and which is 
of such dimensions as to extend far beyond the solar orb.* 

A total eclipse of the sun is one of the most sublime and impres- 
sive phenomena of nature. Among barbarous tribes it is ever con- 
templated with fear and astonishment, while among cultivated na- 
tions it is recognized, from the exactness with which the time of 
occurrence and the various appearances answer to the prediction, 
as affording one of the proudest triumphs of astronomy. By 
astronomers themselves it is of course viewed with the highest 
interest, not only as verifying their calculations, but as contribu- 
ting to establish beyond all doubt the certainty of those grand 
laws, the truth of which is involved in the result. During the 
eclipse of June, 1806, which was one of the most remarkable on 

* See an excellent description and delineation of this appearance as it waf exhibited 
in the eclipse of 1806, in the Transactions of the Albany Institute, by the late Chan- 
eellor De Witt, 



LONGITUDE. 161 

record, the time of total darkness, as seen by the -author of this 
work, was about mid-day. The sky was entirely cloudless, but 
as the period of total obscuration approached, a gloom pervaded 
all nature. When the sun was wholly lost sight of, planets and 
stars came into view ; a fearful pall hung upon the sky, unlike 
both to night and to twilight ; and, the temperature of the air rap- 
idly declining, a sudden chill came over the earth. Even the ani- 
mal tribes exhibited tokens of fear and agitation. 

From 1831 to 1838 w r as a period remarkable for great eclipses 
of the sun, in which time there were no less than five of the most 
remarkable character. The next total eclipse of the sun, visible 
in the United States, will occur on the 7th of August, 1869. 



CHAPTER VIII. 

LONGITUDE TIDES. 

272. As eclipses of the sun afford one of the most approved 
methods of finding the longitudes of places, our attention is natu- 
rally turned next towards that subject. 

The ancients studied astronomy in order that they might read 
their destinies in the stars : the moderns, that they may securely 
navigate the ocean. A large portion of the refined labors of 
modern astronomy, has been directed towards perfecting the as- 
tronomical tables with the view of finding the longitude at sea, 
an object manifestly worthy of the highest efforts of science, con- 
sidering the vast amount of property and of human life involved 
in navigation. 

273. The difference of longitude between two places may be found 
by anty method, by which we can ascertain the difference of their local 
times, at the same instant of absolute time. 

As the earth turns on its axis from west to east, any place that 
lies eastward of another will come sooner under the sun, or will 



162 THE MOON. 

have the sun earlier on the meridian, and consequently, in respect 
to the" hour of the day, will be in advance of the other at the 
rate of one hour for every 15, or four minutes of time for each 
degree. Thus, to a place 15 east of Greenwich, it is 1 o'clock, 
P. M. when it is noon at Greenwich; and to a place 15 west of 
that meridian, it is 11 o'clock, A. M. at the same instant. Hence, 
the difference of time at any two places, indicates their difference 
of longitude. 

* 274. The easiest method of finding the longitude is by means 
of an accurate time piece, or chronometer. Let us set out from 
London with a chronometer accurately adjusted to Greenwich 
time, and travel eastward to a certain place, where the time is 
accurately kept, or may be ascertained by observation. We find, 
for example, that it Is 1 o'clock by our chronometer, when it is 
2 o'clock and 30 minutes at the place of observation. Hence, 
the longitude is 15x1.5=22^ E. Had we travelled westward 
until our chronometer was an hour and a half in advance of the 
time at the place of observation, (that is, so much later in the 
day,) our longitude would have been 22| W. But it would not 
be necessary to repair to London in order to set our chronometer 
to Greenwich time. This might be done at any observatory, or 
any place whose longitude had been accurately determined. For 
example, the time at New York is 4h. 56m. 4 8 .5 behind that of 
Greenwich. If, therefore, we set our chronometer so much be- 
fore the true time at New York, it will indicate the time at Green- 
wich. Moreover, on arriving at different places, any where on 
the earth,, whose longitude is accurately known, we may learn 
whether our chronometer keeps accurate time or not, and if not, 
the amount of its error. Chronometers have been constructed of 
such an astonishing degree of accuracy, as to deviate but a few 
seconds in a voyage from London to Baffin's Bay and back, during 
an absence of several years. But chronometers which are suffi- 
ciently accurate to be depended on for long voyages, are too ex- 
pensive for general use, and the means of verifying their accuracy 
are not sufficiently easy. Moreover, chronometers by being trans- 
ported from one place to another, change 'their daily rate, or de- 
part from that mean rate which they preserve at a fixed station. 



LONGITUDE. 163 

A chronometer, therefore, cannot be relied on for determining the 
longitudes of places where the greatest degree of accuracy is re- 
quired, especially where the instrument is conveyed over land, 
although the uncertainty attendant on one instrument may be 
nearly obviated by employing several and taking their mean 
results.* 

275. Eclipses of the sun and moon are sometimes used for de- 
termining the longitude. The exact instant of immersion or of 
emersion, or any other definite moment of the eclipse which pre- 
sents itself to two distant observers, affords the means of com- 
paring their difference of time, and hence of determining their 
difference of longitude. Since the entrance of the moon into 
the earth's shadow, in a lunar eclipse, is seen at the same instant 
of absolute time at all places where the eclipse is visible, (Art. 
262,) this observation would be a very suitable one for finding 
the longitude were it not that, on account of the increasing dark- 
ness of the penumbra near the boundaries of the shadow, it is 
difficult to determine the precise instant when the moon enters the 
shadow. By taking observations on the immersions of known 
spots on the lunar disk, a mean result may be obtained which will 
give the longitude with tolerable accuracy. In an eclipse of the 
sun, the instants of immersion and emersion may be observed with 
greater accuracy, although, since these do not take place at the 
same instant of absolute time, the calculation of the longitude from 
observations on a solar eclipse are complicated and laborious. 

A method very similar to the foregoing, by observations on 
eclipses of Jupiter's satellites, and on occultations of stars, will 
be mentioned hereafter. 

276. The Lunar method of finding the longitude, at sea, is in J 
many respects preferable to every other. It consists in measuring 
(with a sextant) the angular distance between the moon and the 
sun, or between the moon and a star, and then turning to the Nau- 
tical Almanac,! and finding what time it was at Greenwich when 

* Woodhouse, p. 838. 

t The Nautical Almanac is a book published annually by the British Board of 
ujngitude, containing various tables and astronomical information for the use ol 



164 THE MOON. 

that distance was the same. The moon moves so rapidly, that this 
distance will not be the same except at very nearly the same in- 
stant of absolute time. For example, at 9 o'clock, A. M., at a cer- 
tain place, we find the angular distance of the moon and the sun to 
be 72 ; and on looking into the Nautical Almanac, we find that 
at the time when this distance was the same for the meridian of 
Greenwich was 1 o'clock, P. M. ; hence we infer that the longi- 
tude of the place is four hours, or 60 west. 

The Nautical Almanac contains the true angular distance of 
the moon from the sun, from the four large planets, (Venus, Mars, 
Jupiter, and Saturn,) and from nine bright fixed stars, for the be- 
ginning of every third hour of mean time for the meridian of 
Greenwich ; and the mean time corresponding to any intermediate 
hour, may be found by proportional parts.* 

^M^ <UiL<j^*fik *&* 

277. It would be a very simple operation to determine the lon- 
gitude by Lunar Distances, if the process as described in the 
preceding article were all that is necessary ; but the various cir- 
cumstances of parallax, refraction, and dip of the horizon, would 
differ more or less at the two places, even were the bodies whose 
distances were taken in view from both, which is not necessarily 
the case. The observations, therefore, require to be reduced to 
the center of the earth, being cleared of the effects of parallax and 
refraction. Hence, three observers are necessary in order to take 
a lunar distance in the most exact manner, viz. two to measure 
the altitudes of the two bodies respectively, at the same time that 
the third takes the angular distance between them. The altitudes 
of the two luminaries at the time of observation must be known, 
in order to estimate the effects of parallax and refraction. 

278. Although the lunar method of finding the longitude at 
sea has many advantages over the other methods in use, yet it 

navigators. The American Almanac also contains a variety of astronomical informa- 
tion, peculiarly interesting to the people of the United States, in connexion with a 
vast amount of statistical matter. It is well deserving a place in the library of the 
student. 

* See Bowditch's Navigator, Tenth Ed. p. 226. 



TIDES. 165 

has also its disadvantages. One is, the great exactness requisite 
in observing the distance of the moon from the sun or star, as a 
small error in the distance makes a considerable error in the longi- 
tude. The moon moves at the rate of about a degree in two 
hours, or one minute of space in two minutes of time. There- 
fore, if we make an error of one minute in observing the distance, 
we make an error of two minutes in time, or 30 miles of longitude 
at the equator. A single observation with the best sextants, may 
be liable to an error of more than half a minute ; but the accuracy 
of the result may be much increased by a mean of several obser- 
vations taken to.the east and west of the moon. The imperfection 
of lunar tables was until recently considered as an objection to this 
method. Until within a few years, the best lunar tables were 
frequently erroneous to the amount of one minute, occasioning an 
error of 30 miles. The error of the best tables now in use will 
rarely exceed 7 or 8 seconds.* 

/ 

TIDES. 

279. The tides are an alternate rising and falling of the waters 
of the ocean, at regular intervals. They have a maximum and a 
minimum twice a day, twice a month, and twice a year. Of the 
daily tide, the maximum is called High tide, and the minimum 
Low tide. The maximum for the month is called Spring tide, and 
the minimum Neap tide. The rising of the tide is called Flood 
and its falling Ebb tide. 

Similar tides, whether high or low, occur on opposite sides of 
the earth at once. Thus at the same time it is high tide at any 
given place, it is also high tide on the inferior meridian, and the 
same is true of the low tides. 

The interval between two successive high tides is 12h. 25m. ; 
or, if the same tide be considered as returning to the meridian, 
after having gone around the globe, its return is about 50 minutes 
later than it occurred on the preceding day. In this respect, as 
well as in various others, it corresponds very nearly to the motions 
of the moon. 

* Brinkley's Elements of Astronomy, p. 241. 



166 THE MOON. 

The average height for the whole globe is about 2 feet ; or, 
if the earth were covered uniformly with a stratum of water, the 
difference between the two diameters of the oval would be 5 feet, 
or more exactly 5 feet and 8 inches ; but its natural height at 
various places is very various, sometimes rising to 60 or 70 feet, 
and sometimes being scarcely perceptible. At the same place 
also the phenomena of the tides are very different at different 
times. 

Inland lakes and seas, even those of the largest class, as Lake 
Superior, or the Caspian, have no perceptible tide. 

280. Tidjzs are caused by the unequal attraction of the sun and 
moon upon different parts of the earth. 

Suppose the projectile force by which the earth is carried for- 
ward in her orbit, to be suspended, and the earth to fall towards 
one of these bodies, the moon, for example, in consequence of 
their mutual attraction. Then, if all parts of the earth fell 
equally towards the moon, no derangement of its different parts 
would result, any more than of the particles of a drop of water 
in its descent to the ground. But if one part fell faster than an- 
other, the different portions would evidently be separated from 
each other. Now this is precisely what takes place with respect 
to the earth in its fall towards the moon. The portions of the 
earth in the hemisphere next to the moon, on account of being 
nearer to the center of attraction, fall faster than those in the op- 
posite hemisphere, and consequently leave them behind. The 
solid earth, on account of its cohesion, cannot obey this impulse, 
since all its different portions constitute one mass, which is acted 
on in the same manner as though it were all collected in the cen- 
ter : but the waters on the surface, moving freely under this im- 
pulse, endeavor to desert the solid mass and fall towards the 
moon. For a similar reason the waters in the opposite hemisphere 
falling less towards the moon than the solid earth, are left behind, 
or appear to rise from the center of the earth. 

281. Let DEFG (Fig. 56,) represent the globe ; and, for the sake 
of illustrating the principle, we will suppose the waters entirely to 
cover the globe at a uniform depth. Let defg repf esent the solid 



TIDES. 



167 




globe, and the circular ring exterior to Fig. 56. 

it, the covering of waters. Let C be 
the center of gravity of the solid mass, 
A that of the hemisphere next to the 
moon, and B that of the remoter hemi- 
sphere. Now the force of attraction 
exerted by the moon, "acts in the same 
manner as though the solid mass were 
all concentrated in C, and the waters 
of each hemisphere at A and B respec- 
tively ; and (the moon being supposed above E) it is evident that 
A will tend to leave C, and C to leave B behind. The same must 
evidently be true of the respective portions of matter, of which 
these points are the centers of gravity. The waters of the globe 
will thus be reduced to an oval shape, being elongated in the direc- 
tion of that meridian which is under the moon, and flattened in 
the intermediate parts, and most of all at points ninety degrees dis- 
tant from that meridian. 

Were it not, therefore, for impediments which prevent the force 
from producing its full effects, we might expect to see the great 
tide- wave, as the elevated crest is called, always directly beneath 
the moon, attending it regularly around the globe. But the in- 
ertia of the waters prevents their instantly obeying the moon's 
attraction, and the friction of the waters on the bottom of the 
ocean, still further retards its progress. It is not therefore until 
several hours (differing at different places) after the moon has 
passed the meridian of a place, that it is high tide at that place. 

282. The sun has a similar action to the moon, but only one 
third as great. On account of the great mass of the sun. com- 
pared with that of the moon, we might suppose that his action 
in raising the tides would be greater than the moon's ; but the 
nearness of the moon to the earth more than compensates for 
the sun's greater quantity of matter. Let us, however, form a 
just conception of the advantage which the moon derives from her 
proximity. It is not that her actual amount of attraction is thus 
rendered greater than that of the sun ; but it is that her attraction 
for the different parts of the earth is very unequal, while that of 



168 THE MOON. 

the sun is nearly uniform. It is the inequality of this action, and 
not the absolute force, that produces the tides. The diameter of 
the earth is V f the distance of the moon, while it is less than 
reioT of the distance of the sun. 

283. Having now learned the general cause of the tides, we 
will next attend to the explanation of particular phenomena. 

The Spring tides, or those which rise to an unusual height 
twice a month, are produced by the sun and moon's acting to- 
gether; and the Neap tides, or those which are unusually low 
twice a month, are produced by the sun and moon's acting in 
opposition to each other. The Spring tides occur at the syzygies ; 
the Neap tides at the quadratures. At the time of new moon, 
the sun and moon both being on the same side of the earth, and 
acting upon it in the same line, their actions conspire, and the 
sun may be considered as adding so much to the force of the 
moon. We have already explained how the moon contributes to 
raise a tide on the opposite side of the earth. But the sun as well 
as the moon raises its own tide-wave, which, at new moon, coin- 
cides with the lunar tide-wave. At full moon, also, the two lumina- 
ries conspire in the same way to raise the tide ; for we must recol- 
lect that each body contributes to raise the tide on the opposite 
side of the earth as well as on the side nearest to it. At both the 
conjunctions and oppositions, therefore, that is, at the syzygies, 
we have unusually high tides. But here also the maximum effect 
is not at the moment of the syzygies, but 36 hours afterwards. 

At the quadratures, the solar wave is lowest where the lunar 
wave is highest ; hence the low tide produced by the sun is sub- 
tracted from high water and produces the Neap tides. Moreover, 
at the quadratures the solar wave is highest where the lunar wave 
is lowest, and hence is to be added to the height of low water at 
the time of Neap tides. Hence the difference between high and 
low water is only about half as great at Neap tide as at Spring tide. 

284. The power of the moon or of the sun to raise the tide is 
found by the doctrine of universal gravitation to be inversely as 
the cube of the distance* The variations of distance in the sun are 

* La Place, Syst. du Monde, 1. iv, c. x. 






TIDES. 



169 



not great enough to influence the tides very materially, but the 
variations in the moon's distances have a striking effect. The 
tides which happen when the moon is in perigee, are considerably 
greater than when she is in apogee ; and if she happens to be in 
perigee at the time of the syzygies, the spring tide is unusually 
high. When this happens at the equinoxes, the highest tides of 
the year are produced. 

285. The declinations of the sun and moon have a considerable 
influence on the height of the tide. When the moon, for example, 
has no declination, or is in the equator, as in figure 57,* the rota- 
tion of the earth on its axis NS will make the two tides exactly 
equal on opposite sides of the earth. Thus a place which is car- 
ried through the parallel TT' will have the height of one tide T2 
and the other tide T'3. The tides are in this case greatest at the 
equator, and diminish gradually to the poles, where it will be low 
water during the whole day. When the moon is on the north side 
of the equator, as in figure 58, at her greatest northern declination, 

Fig. 58. 

3S" x-^ 





a place describing the parallel TT' will have T'3 for the height of 
the tide when the moon is on the superior meridian, and T2 for 
the height when the moon is on the inferior meridian. Therefore, 
all places north of the equator will have the highest tide when the 
moon is above the horizon, and the lowest when she is below it ; 
the difference of the tides diminishing towards the equator, where 

* Diagrams like these are apt to mislead the learner, by exhibiting the protuberance 
occasioned by the tides as much greater than the reality. We must recollect that it 
amounts, at the highest, to only a very few feet in eight thousand miles. Were the 
diagram, therefore, drawn in just proportions, the alterations of figure produced by the 
lides would be wholly insensible. 

22 



170 THE MOON. 

they are equal. In like manner, places south of the equator have 
the highest tides when the moon is below the horizon, and the 
lowest when she is above it. When the moon is at her greatest 
declination, the highest tides will take place towards the tropics. 
The circumstances are all reversed when the moon is south of the 
equator.* 

286. The motion of the tide-wave, it should be remarked, is not 
a progressive motion, but a mere undulation, and is to be carefully 
distinguished from the currents to which it gives rise. If the 
ocean completely covered the earth, the sun and moon being in the 
equator, the tide-wave would travel at the same rate as the earth 
on its axis. Indeed, the correct way of conceiving of the tide- 
wave, is to consider the moon at rest, and the earth in its rotation 
from west to east as bringing successive portions of water under 
the moon, which portions being elevated successively at the same 
rate as the earth revolves on its axis, have a relative motion west- 
ward in the same degree. 

287. The tides of rivers, narrow bays, and shores far from the 
main body of the ocean, are not produced in those places by the 
direct action of the sun and moon, but are subordinate waves 
propagated from the great tide-wave. 

Lines drawn through all the adjacent parts of any tract of wa- 
ter, which have high water at the same time, are called cotidal 
lines.-\ We may, for instance, draw a line through all places in 
the Atlantic Ocean which have high tide on a given day at 1 o'clock, 
and another through all places which have high tide at 2 o'clock. 
The cotidal line for any hour may be considered as representing 
the summit or ridge of the tide-wave at that time ; and could the 
spectator, detached from the earth, perceive the summit of the 
wave, he would see it travelling round the earth in the open ocean 
once in twenty four hours, followed by another twelve hours dis- 
tant, and both sending branches into rivers, bays, and other open- 
ings into the main land. These latter are called Derivative tides, 



* Edm. Encyc. Art. Astronomy, p. 623. 

t Whewell, Phil. Transaction for 1833, p. 148. 



TIDES. 



171 



while those raised directly oy the action of the sun and moon, are 
called Primitive tides. 

288. The velocity with which the wave moves will depend on 
various circumstances, but principally on the depth, and probably 
on the regularity of the channel. If the depth be nearly uniform, 
the cotidal lines will be nearly straight and parallel. But if some 
parts of the channel are deep while others are shallow, the tide 
will be detained by the greater friction of the shallow places, and 
the cotidal lines will be irregular. The direction also of the de- 
rivative tide, may be totally different from that of the primitive. 
Thus, (Fig. 59,) if the great tide- Fig. 59. 

wave, moving from east to west, 
be represented by the lines 1, 2, 
3, 4, the derivative tide which is 
propagated up a river or bay, 
will be represented by the cotidal 
lines 3, 4, 5, 6, 7. Advancing 
faster in the channel than next 
the banks, the tides will lag be- 
hind towards the shores, and the 
cotidal lines will take the form 
of curves as represented in the 
diagram. 

289. On account of the retarding influence of shoals, and an 
uneven, indented coast, the tide-wave travels more slowly along 
the shores of an island than in the neighboring sea, assuming con- 
vex figures at a little distance from the island and on opposite 
sides of it. These convex lines sometimes meet and become 
blended in such a manner as to create singular anomalies in a sea 
much broken by islands, as well as on coasts indented with numer- 
ous bays and rivers.* Peculiar phenomena are also produced, 
when the tide flows in at opposite extremities of a reef or island, 
as into the two opposite ends of Long Island Sound. In certain 




* See an excellent representation and description of these different phenomena by 
Professor Whewell, Phil. Trans. 1833, p. 153. 



172 THE MOON. 

cases a tide-wave is forced into a narrow arm of the sea, and 
produces very remarkable tides. The tides of the Bay of Fundy 
(the highest in the world) sometimes rise to the height of 60 or 70 
feet ; and the tides of the river Severn, near Bristol in England, 
rise to the height of 40 feet. 

290. The Unit of Altitude of any place, is the height of the 
maximum tide after the syzygies, (Art. 283,) being usually about 
36 hours after the new or full moon. But as the amount of this 
tide would be affected by the distance of the sun and moon from 
the earth, (Art. 284,) and by their declinations, (Art. 285,) these 
distances are taken at their mean value, and the luminaries are 
supposed to be in the equator ; the observations being so reduced 
as to conform to these circumstances. The unit of altitude can be 
ascertained by observation only. The actual rise of the tide de- 
pends much on the strength and direction of the wind. When 
high winds conspire with a high flood tide, as is frequently the 
case near the equinoxes, the tide rises to a very unusual height. 
We subjoin from the American Almanac a few examples of the 
unit of altitude for different places. 

Feet. 

Cumberland, head of the Bay of Fundy, 71 
Boston, . . . . .Ill 

New Haven, .... 8 

New York, .... 5 

Charleston, S. C., . .6 

291. The Establishment of any port is the mean interval between 
noon and the time of high water, on the day of new or full moon. 
As the interval for any given place is always nearly the same, it 
becomes a criterion of the retardation of the tides at that place. 
On account of the importance to navigation of a correct know- 
ledge of the tides, the British Board of Admiralty, at the sugges- 
tion of the Royal Society, recently issued orders to their agents 
in various important naval stations, to have accurate observations 
made on the tides, with the view of ascertaining the establishment 
and various other particulars respecting each station;* and the 

* Lubbock, Report on the Tides, 1833. 



TIDES. 173 

government of the United States is prosecuting similar investiga- 
tions respecting our own p.orts. 

292. According to Professor Whewell,* the tides on the coast 
of North America are derived from the great tide-wave of the 
South Atlantic, which runs steadily northward along the coast to 
the mouth of the Bay of Fundy, where it meets the northern tide 
wave flowing in the opposite direction. Hence he accounts for 
the high tides of the Bay of Fundy. 

293. The largest lakes and inland seas have no perceptible 
tides. This is asserted by all writers respecting the Caspian and 
Euxine, and the same is found to be true of the largest of the 
North American lakes, Lake Superior, f 

Although these several tracts of water appear large when taken 
by themselves, yet they occupy but small portions of the surface 
of the globe, as will appear evident from the delineation of them 
on an artificial globe. Now we must recollect that the primitive 
tides are produced by the unequal action of the sun and moon 
upon the different parts of the earth ; and that it is only at points 
whose distance from each other bears a considerable ratio to the 
whole distance of the sun or the moon, that the inequality of ac- 
tion becomes manifest. The space required is larger than either 
of these tracts of water. It is obvious also that they have no op- 
portunity to be subject to a derivative tide. 

294. To apply the theory of universal gravitation to all the va- 
rying circumstances that influence the tides, becomes a matter of 
such intricacy, that La Place pronounces "the problem of the 
tides" the most difficult problem of celestial mechanics. 

295. The Atmosphere that envelops the earth, must evidently be 
subject to the action of the same forces as the covering of waters, 
and hence we might expect a rise and fall of the barometer, indi- 
cating an atmospheric tide corresponding to the tide of the ocean. 



* Phil. Trans. 1833, p. 172. 

t See Experiments of Gov. Cass, Am. Jour. Science. 



174 THE PLANETS. 

La Place has calculated the amount of this aerial tide. It is too 
inconsiderable to be detected by changes in the barometer, unless 
by the most refined observations. Hence it is concluded, that 
the fluctuations produced by this cause are too slight to affect 
meteorological phenomena in any appreciable degree.* 



CHAPTER IX. 

OF THE PLANETS INFERIOR PLANETS, MERCURY AND VENUS. 

296. THE name planet signifies a wanderer,^ and is applied to 
this class of bodies because they shift their positions in the 
heavens, whereas the fixed stars apparently always maintain the 
same places with respect to each other. The planets known 
from a high antiquity, are Mercury, Venus, Earth, Mars, Jupi- 
ter, and Saturn. To these, in 1781, was added Uranus, J (or 
Herschel, as it was formerly called, from the name of its discov- 
erer,) and, as late as 1846, another large planet, Neptune, was 
added to the list, making eight in all of the regular series. Be- 
sides these, there are found between Mars and Jupiter, a remarka- 
ble group of small planets, called Asteroids, numbering at present 
thirty-three. Of these, four Ceres, Pallas, Juno, and Yesta 
were discovered near the commencement of the present century ; 
and the remaining twenty-nine, Astrea, Hebe, Iris, Flora, Metis, 
Hygeia, Parthenope, Yictoria, Egeria, Irene, Eunomia, Psyche, 
Thetis, Melpomene, Fortuna, Massalia, Lutetia, Calliope, Thalia, 
Themis, Phocea, Proserpina, Euterpe, Bellona, Amphytrite, 
Urania, Euphrosyne, Pomona, and Polymnia, have been dis- 
covered since the year 1845. 

The foregoing are called primary planets. Several of these 
have one or more attendants, or satellites, which revolve around 
them as they revolve around the sun. The Earth has one sat- 
ellite, namely, the moon ; Jupiter has four ; Saturn, eight ; Ura- 

* Bowditch'a La Place, II. 797. f From the Greek, irAaj/^j. 

From Ovpavos. 



DISTANCES FROM THE SUN. 175 

nus, six ;* and Neptune, one. These bodies also are planets, but 
in distinction from the others they are called secondary planets. 
It appears, therefore, that the whole number of planets known 
at present are 54, viz., 8 primary, 20 secondary, and 26 
asteroids. 

297. The primary planets all (with the exception of the as- 
teroids) have their orbits nearly in the same plane, and are never 
seen far from the ecliptic. Mercury, whose orbit is most inclined 
of all, never departs further from the ecliptic than about 7, while 
most of the other planets pursue very nearly the same path with 
the earth, in their annual revolution around the sun. The aste- 
roids, however, make wider excursions from the plane of the 
ecliptic, amounting, in the case of Pallas, to 34j. 

298. Mercury and Venus are called inferior planets, because 
their orbits are nearer to the sun than that of the earth ; while 
all the others being more distant from the sun than the earth, 
are called superior planets. The planets present great diver- 
sities among themselves in respect to distance from the sun, mag- 
nitude, time of revolution, and density. They differ also in 
regard to satellites, of which, as we have seen, the Earth and 
Neptune have each one, Jupiter has four, Saturn eight, and 
Uranus six ; while Mercury, Venus, and Mars, have none at all. 
It will aid the memory, and render our view of the planetary 
system more clear and comprehensive, if we classify, as far as 
possible, the various particulars comprehended under the fore- 
going heads. 

299. DISTANCES FROM THE SUN.f 

1. Mercury, 37,000,000 0.3870981 

2. Venus, 68,000,000 0.7233316 



* Respecting the number of satellites belonging to Uranus, there is some doubt, 
which will be considered under the history of that planet. 

f The distances in miles, as expressed in the first column, are to be treasured 
up in the memory, while the second column expresses the relative distance, that of 
the Earth being 1, from which a more exact determination may be made when re- 
quired, the Earth's distance being taken at 9^,298,260 miles. 



176 THE PLANETS. 



3. Earth, 


95,000,000 


1.0000000 


4. Mars, 


145.000,000 


1.5236923 


5. Asteroids, 


250,000,000 


2.6612885 


6. Jupiter, 

7. Saturn, 


495,000,000 , 
900,000,000 


5.2027760 
9.5387861 


8. Uranus, 


1,800,000,000 


19.1823900 


9. Neptune, 


2,800,000,000 


30.0318000 



The dimensions of the planetary system are seen from this 
table to be vast, comprehending a circular space nearly six 
thousand millions of miles in diameter. A railway car, travelling 
night and day at the rate of 20 miles an hour, and of course 
making 480 miles a day, would require about 50 days to travel 
round the Earth on a great circle, and about 500 days to reach 
the moon ; but it will give some idea of the vastness of the 
planetary spaces to reflect, that setting out from the sun, and 
travelling from planet to planet at the same rate, to reach Mer- 
cury would require about 200 years ; Venus, nearly 400 ; the 
Earth, 54 ; Mars, more than 800 ; Jupiter, towards 3,000 ; Sat- 
urn, above 5,000; Uranus, 10,000 ; Neptune, more than 16,000 ; 
and to cross the entire orbit of Neptune would require upwards 
of 32,000 years. 

It may aid the memory to remark, that in regard to the plan- 
ets nearest the sun, the distances increase in an arithmetical 
ratio, while those most remote increase in a geometrical ratio. 
Thus, if we add 30 to the distance of Mercury, it gives us 
nearly that of Venus ; 30 more gives that of the Earth ; while 
Saturn is nearly twice the distance of Jupiter, and Uranus twice 
that of Saturn. If this, however, were a perfectly correct rule, 
Neptune would be twice as far from the sun as Uranus, and 
therefore 3,600 millions of miles, whereas its actual distance is 
short of 3,000 millions. Between the orbits of Mars and Jupi- 
ter a great chasm appeared, which broke the continuity ; but 
the discovery of the Asteroids has filled the void. A more exact 
law of the series is that called Bode's law. w It is as follows : if 
we represent the distance of Mercury by 4, and increase the fol- 
lowing terms by the product of 3 into the ascending powers of 2, 
we shall obtain the relative distances of the planets from the 
sun. Thus, 



MAGNITUDES 177 

Mercury, .... 4 =4. 

Venus, .... 4 + 3.2 / = 7 
Earth, .... 4 + 3.2 1 1 =10 
Mars, .... 4 + 3.2 2 - = 16 ^ 
Ceres, .... 4+3.2 3 * = 28 * 
Jupiter, .... 4+3.2* ib = 52 
Saturn, .... 4+3.2* SZ. =100 
Uranus, .... 4+3.2 6 , :? - =196 
Neptune,. . . . 4+3.2^1^=388 
For example, by this law, the distances of the earth and Jupi- 
ter are to each other as 10 to 52. Their actual distances, as 
given in the table, (Art. 299,) are as 1 to 5.202776, which num- 
bers are nearly as 10 to 52. 

The mean distances of the planets from the sun, may also be 
determined by Kepler's law, that the squares of the periodic 
times are as the cubes of the distances, (Art. 192.) Thus the 
earth's distance being previously ascertained by means of the 
sun's horizontal parallax, (Art. 87,) and the period of any other 
planet as Jupiter, being learned from observation, we may say as 
the square of the earth's period (365.256 days) is to the square 
of Jupiter's period, (433&5S6 days,) so is the cube of 1 -year to 
the cube of Jupiter's pewSSfthe cube root of which will be the 
period itself. Or, to express the same truth more concisely, 
365.256 2 : 4332.5S6 2 : : 1 s : 5.202 3 . 

300. MAGNITUDES. 





Diameter 
in Miles. 


Mean apparent 
Diameter. 


Mass. 


Volume. 


Mercury, . 


2950* 


8". 


4,865,751f 


A 


Venus, 


. 7800 


17". 


401,839 


T 9 o 


Earth, 


7912 




389,551 


1 


Mars, 


4500 


6". 


2,680,337 


t 


Ceres, 


160 


0".5 






Jupiter, . 


. 89000 


37". 


1,048 


1400 


Saturn, 


. 79000 


16'. 


3,502 


1000* 


Uranus 


' . . 35000 


4 . . 


24,905 


86 


Neptune . 


. 31000* 


2".5 


18,780 


60 



* Hind. 

f Herschel. The numbers of this column express the denominators of frac- 
tions, of which the numerator is 1, denoting the sun's mass. 

23 



178 THE PLANETS. 

Diagrams and orreries, as usually constructed, wholly lail of 
giving any just conceptions of the distances of the planets from 
the sun and from each other. If we represent, for instance, the 
distance of the earth by 1 foot, we shall require 30 feet in ordei 
to reach the place of Neptune ; and when we have constructed 
a diagram on so enlarged a scale, we must still bear in mind that 
each foot represents a space of nearly 100 millions of miles.* 

We remark here a great diversity in regard to magnitude a 
diversity which does not appear to be subject to any definite 
law. While Venus, an inferior planet, is nine-tenths as large 
as the earth, Mars, a superior planet, is only one-sixth, while 
Jupiter is fourteen hundred times -as large. Although sev- 
eral of the planets, when nearest to us, appear brilliant and 
large when compared with the fixed stars, yet the angle which 
they subtend is very small, that of Venus, the greatest of all, 
never exceeding about 1', or more exactly 61 ".2, and that of 
Jupiter, when greatest, being only about f of a minute. 

The distance of one of the near planets, as Venus or Mars, 
may be determined from its parallax ; and the distance being 
known, its real diameter can be estimated from its apparent 
diameter, in the same manner as we estimate the diameter of the 
sun. (Art. 145.) 



301. PERIODIC TIMES. 

Sidereal revolution. /*" 


Mean daily motion. 


Mercury, 3 months, or / 88 days, 


4 5' 32".6 


Venus, 7J " 


224 " 


1 36' 7".8 


Earth, 1 year, 


365 " 


59' 8".3 


Mars, 2 " 


687 " 


31' 26".7 


Ceres, 4} " 


1687 " 


12' 50".9 


Jupiter, 12 


4332 " 


4' 59".3 


Saturn, 29 


10759 " 


2' 0".6 


Uranus, 84 


30686 " 


0' 42".4 


Neptune, 164J " 


" 60127 " 


0' 21". 5 



* For the purposes of illustration to a class or to a popular audience, the folio-w- 
ing plan of representation is recommended, not only for the entire solar system, but 
for each of the subordinate systems, as that of Jupiter or of Saturn. Procure a 
few sheets of black paper ; cut it into strips a foot wide, and paste them to- 



INFERIOR PLANETS MERCURY AND VENUS. 179 

From this view it appears that the planets nearest the sun 
move most rapidly. Thus Mercury performs nearly 350 revolu- 
tions while Uranus performs one. This is evidently not owing 
merely to the greater dimensions of the orbit of Uranus, for the 
length of its orbit is not 50 times that of the orbit of Mercury, 
while the time employed in describing it is 350 times that of 
Mercury. Indeed, this ought to follow from Kepler's law, that 
the squares of the periodic times are as the cubes of the distan- 
ces ; from which it is manifest that the times of revolution in- 
crease faster than the dimensions of the orbit. Accordingly, the 
apparent progress of the most distant planets is exceedingly slow, 
the rate of Uranus being only 42" A per day ; so that for weeks 
and months, and even years, this planet but slightly changes its 
place among the stars. 

The planets are divided into two classes, first, the inferior, 
which have their orbits nearer to the sun than that of the earth ; 
and secondly, the superior, which have their orbits exterior to 
the earth's orbit. 

THE INFERIOR PLANETS, MERCURY AND VENUS. 

302. The inferior planets, Mercury and Venus, having their 
orbits far within that of the earth, appear to us as attendants 
upon the sun. Mercury never appears further from the sun than 
29, (28 48',) and seldom so far ; and Venus never more than 
about 47, (47 12'.) Both planets, therefore, appear either in 
the west soon after sunset, or in the east a little before sunrise. 
In high latitudes, where the twilight is prolonged, Mercury can 
seldom be seen with the naked eye, and then only at the periods 
of its greatest elongation.* The reason of this will readily ap- 
pear from the following diagram. 



gether, so as to form a continuous sheet. For the solar system, this may be about 
30 feet long. Cut out of white paper figures representing the sun and each of me 
planets, (and, if desired, each of the satellites,) which paste on the long sheet at 
distances corresponding to their respective ratios, that of the Earth being 1. This 
enlarged diagram may be exhibited on a wall, or on a base made of boards ex- 
tended along in a line with each other, and hung upon a wall. 

* Copernicus is said to have lamented on his death-bed that he had never been 
able to obtain a sight of Mercury, and Delambre, a great French astronomer, saw 
it but twice. 



180 THE PLANETS. 

Let S represent the sun, E the earth, and MN Mercury at 
its greatest elongations from the sun, and OZP a portion of the 
sky. Then, since we refer all distant bodies to the same concave 
sphere of the heavens, we should see the sun at Z, and Mercury 
at O, when at its greatest eastern elongation, and at P when at 
its greatest western elongation ; and while passing from M to N 
through Q, it would appear to describe the arc OP ; and while 
passing from N to M through R, it would appear to run back 
across the sun on the same arc. It is further evident, that it 
would be visible only when at or near one of its greatest elonga- 
tions ; being at all other times so near the sun as to be lost in 
his light. 




E 



303. A planet is said to be in conjunction with the sun, when 
it is seen in the same part of the heavens with the sun, or 
when it has the same longitude. Mercury and Venus have each 
two conjunctions, the inferior and the superior. The inferior 
conjunction is its position when in conjunction on the same side 
of the sun with the earth, as at Q in the figure : the superior 
conjunction is its position when on the side of the sun most dis- 
tant from the earth, as at R. 

304. The period occupied by a planet between two successive 
conjunctions with the earth, is called its synodical revolution. 
Both the planet and the earth being in motion, the time of the 



INFERIOR PLANETS MERCURY AND VENUS. 181 

synodical revolution exceeds that of the sidereal revolution of 
Mercury or Venus ; for when the planet comes round to the 
place where it before overtook the earth, it does not find the 
Earth at that point, but far in advance of it. Thus, let Mercury 
come into conjunction with the earth at Q, (Fig 60.) In about 
88 days the planet will come round to the same point again ; 
but meanwhile the earth has moved forward through nearly a 
fourth part of her revolution, and will continue to move onward 
while Mercury, with a swifter motion, is following on to over- 
take her, the case being analogous to the hour and second-hand 
of a clock. Having the sidereal period of a planet, which may 
always be accurately determined by observation, we may ascer- 
tain its synodical period as follows. 

By the table in article 301, the mean daily motion of Mercury 
is 4 5' 32".6= 14732".6, and that of the earth is 59' 8".3=3548".3. 
Therefore 14732".6-3548".3=11184".3, which is the average 
gain of Mercury over the earth in a day. But in order to over.- 
take the earth, Mercury must complete one revolution, and as 
much of another as the earth has performed until the planet over- 
takes it ; that is, the planet must gain an entire revolution. 
Now, 

11184".3 : 1 day : : 360 : 115.8 days, the synodical period of 
Mercury. In like manner, the daily gain of Venus is 2219". 5, 
and 

22 19". 5 : : 1 day : : 360 : 583.9 days, the synodical period of 
Venus. 

305. The motion of an inferior planet is direct in passing 
through its superior conjunction, and retrograde in passing 
through its inferior conjunction. 

Thus Venus, while going from B through D to A, (Fig. 61,) 
moves in the order of the signs, or from west to east, and would 
appear to traverse the celestial vault B' S' A' from right to left ; 
but in passing from A through C to B, her course would be ret- 
rograde, returning on the same arc from left to right. If the 
earth were at rest, therefore, (and the sun of course at rest,) the 
inferior planets would appear to oscillate backwards and forwards 
across the sun. But, it must be recollected that the Earth is 



182 



THE PLANETS. 



moving in the same direction with the planet, as respects the signs, 
but with a slower motion. This modifies the apparent motions of 
the planet, accelerating it in the superior, and retarding it in the 
inferior conjunctions. Thus, in Figure 61, Venus, while moving 
through BDA, would seem to move in the heavens from B' to A' 
were the earth at rest ; but meanwhile the earth changes its po- 
sition from E to E', by which means the planet is not seen at A' 
but at A", being accelerated by the arc A' A", in consequence 
of the earth's motion. On the other hand, when the planet is 
passing through its inferior conjunction ACB, it would appear to 

Fig. 61. 




move backwards in the heavens A' to B' if the earth were at 
rest, but from A' to B" if the earth has in the mean time moved 
from E to E', being retarded by the arc B'B". Although the 
motions of the earth have the effect to accelerate the planet in 
the superior, and to retard it in the inferior conjunction, yet on 
account of the greater distance, the apparent motion of the 
planet is much slower in the superior than in the inferior con- 
junction. 

306. When passing from the superior to the inferior conjunc- 
tion, or from the inferior to the superior through the points of 
greatest elongation, the inferior planets are stationary. 

If the earth were at rest, the stationary points would be at the 



INFERIOR PLANETS MERCURY AND VENUS. 183 

greatest elongations, as at A and B, for then the planet would be 
moving directly towards or from the earth, and would be seen 
for some time in the same place in "the heavens ; but the earth 
itself is moving nearly at right angles to the line of the planet's 
motion ; hence a direct apparent motion is given to the planet 
by this cause. When the planet, however, has passed this line, 
by its superior velocity it soon overcomes this tendency of the 
earth to give it an apparent motion eastward, and becomes ret- 
rograde as it approaches the inferior conjunction, ffs station- 
ary point evidently lies between its place of greatest elongation, 
and the place where its motion becomes retrograde. Mercury 
is stationary at an elongation from 15 to 20 from the sun, and 
Venus at about 29. The former continues to retrograde during 
22 days ; the latter, about 42.* 

307. Mercury and Venus -exhibit to the telescope phases similar 
to those of the moon. 

When on the side of their inferior conjunction, as from A to 
B through C, (Fig. 61,) these planets appear horned, like the 
moon in her first and last quarters ; and when on the side of 
their superior conjunctions, as from B to A through D, they ap- 
pear gibbous. At the moment of superior conjunction, the 
whole enlightened orb of the planet is turned towards the earth, 
and the appearance would be that of the full moon, but the 
planet is too near the sun to be commonly visible. All these 
changes of figure resulting from the different positions of the 
planet with respect to the sun and earth, will be readily under- 
stood by inspecting the diagram, (Fig. 61.) 

These different phases show that these bodies are opake, and 
shine only as they reflect to us the light of the sun ; and the 
same remark applies to all the planets. 

308. The distance of an inferior planet from the sun, may be 
found by observations at the time of its greatest elongation. 

Thus if E (Fig. 62) be the place of the earth, and C that of 
Venus at the time of her greatest elongation, the angle SCE will be 

* Herschel's Outlines, 278 ; Woodhouse, 657, 




184 THE PLANETS. 

Fig. 62. 

known, being a right angle. Also the angle SEC 
is known from observation. Hence the ratio of 
SC to SE becomes known ; or, since SE is 
given, being the distance of the earth from the 
sun, SC the radius of the orbit of the planet is 
determined. If, therefore, we already know 
the distance of the earth from the sun, we can 
by this problem easily find the distance of Mer- 
cury or Venus ; or, if neither were actually 
known, their ratio to each other would be 
found by this method. If the orbits were 
both circles, this method would be very exact ; 
but being elliptical, we obtain the mean value of the radius 
SC by observing its greatest elongation in different parts of 
its orbit.* 

308. The orbit of Mercury is the most eccentric, and the 
most inclined of all the planets ;f while that of Venus varies but 
little from a circle, and lies much nearer to the ecliptic. 

The eccentricity of the orbit of Mercury is nearly of its semi- 
major axis, while that of Venus is T ^5 ; and that of the earth 
only sV ; the inclination of Mercury's orbit is 7, while that of 
Venus is only 3^4 At the perihelion, Mercury is only 29 mil- 
lions of miles from the sun, while at the aphelion his distance is 
44 millions, a variation of 15 millions, and more than five times 
as great as in the case of the earth. On account of his differ- 
ent distances from the earth, Mercury is also subject to much 
variation in his apparent diameter, which is 12'' in perigee, but 
only 5" in apogee. 

310. The most favorable time for determining the sidereal 
revolution of a planet, is when its conjunction takes place at one 
of its nodes ; for then the sun, the earth, and the planet, being 
in the same straight line, it is referred to its true place in the 



* Herschel's Outlines, p. 275 

f The asteroids are of course excepted. $ Baily's Tables. 



INFERIOR PLANETS MERCURY AND VENUS. 185 

heavens, whereas, in other positions, its apparent place is more 
or less affected by perspective. 

311. An inferior planet is brightest at a certain point be- 
tween its greatest elongation and inferior conjunction. 

Its maximum brilliancy would happen at its inferior conjunc- 
tion, (being then nearest to us,) if it shone by its own light ; but 
in that position its dark side is turned towards us. Still its max- 
imum cannot be when most of the illuminated side is turned 
towards us ; for then, being at the superior conjunction, it is at 
its greatest distance from us. The maximum must, therefore, 
be somewhere between the two. Venus gives her greatest light 
when about 40 from the sun. 

,312. Mercury and Venus both revolve on their axes in nearly 
the same time with the earth. 

The diurnal period of Mercury is a little greater than that 
of the earth, being 24h. 5m. 28s., and that of Venus is a little 
less than the earth's, being 23h. 21m. 7s. The revolutions on 
their axes have been determined by means of some spot or mark 
seen by the telescope, as the revolution of the sun on his axis is 
ascertained by means of his spots.. 

313. Venus is regarded as the most beautiful of the planets, 
and is well known as the morning and evening star. The most 
ancient nations did not indeed recognize the evening and morn- 
ing star as one and the same body, but supposed they were dif- 
ferent planets, and accordingly gave them different names, calling 
the morning star Lucifer, and the evening star Hesperus. At 
her period of greatest splendor, Venus casts a shadow, and is 
sometimes visible in broad daylight. This occurred in a very 
striking mannner in September, ] 852, Venus being on the merid- 
ian about 9 o'clock, A. M., and her northern declination nearly 
15 degrees. Although not 45 from the inferior conjunction, and 
consequently exposing only a portion of her disk, like that of the 
moon when three or four days old, yet her light is then estimated 
as equal to that of twenty stars of the first magnitude.* At her 

* Francosur, Uranography, p. 125. 
24 



186 - THE PLANETS. 

period of greatest elongation, Venus is visible from three to four 
hours after the setting, or before the rising of the sun. 

314. Every eight years, Venus forms her conjunctions with the 
sun in the same part of the heavens. 

The sidereal period of Venus being 224.7 days, and that of the 
earth 365.256 days, thirteen revolutions of Venus are accom- 
plished in nearly the same time as eight revolutions of the earth : 
for 224.7 X 13 = 2921, and 365.256 X 8 = 2922. At the end 
therefore of 2922 days, or eight years, the two bodies will come 
round to the same point of the heavens, and be in the same situ- 
ation with respect to each other, as at the beginning. Conse- 
quently, whatever appearances of this planet arise from its posi- 
tions wi-th respect to the earth and the sun, (as, for example, 
being visible in the daytime,) they are repeated every eight years 
in nearly the same form. 

TRANSITS OF THE INFERIOR PLANETS 

315. The transit of Mercury or Venus, is its passage across 
the suns disk, as the moon passes over it in a solar eclipse. 

As a transit takes place .only when the planet is in inferior 
conjunction, at which time her motion is retrograde, (Art. 305,) 
it is always from left to right, and the planet is seen projected 
on the solar disk in a black round spot. Were the orbits of these 
planets coincident with the earth's orbit, a transit would occur at 
some part of the earth at every inferior conjunction, as there 
would be an eclipse of the sun at every new moon, were the 
moon's revolution in the plane of the ecliptic. But the orbit of 
Venus makes an angle of 3^ with that of the earth, and the orbit 
of Mercury an angle of 7 ; and, moreover, the apparent diame 
ter of each of these bodies is very small, both of which circum. 
stances conspire to render a transit a comparatively rare occur 
rence, since it can happen only when the sun, at the time of an 
inferior conjunction, happens to be at, or extremely near the 
planet's node. The nodes of Mercury lie in that part of the 
earth's orbit which it passes in the months of May and Novem- 
ber. It is only in these months, therefore, that transits of Mer- 
cury can occur. For a similar reason, those of Venus occur only 



TRANSITS OF THE INFERIOR PLANETS. 187 

in June and December. Since the nodes of both planets have a 
small retrograde motion, the months in which transits occur, 
will change in the course of ages ; but the months for transits 
will for a long time remain the same as at present, since the 
nodes of Mercury change their places only in 13', and those of 
Venus only 31' in a century.* 

The first prediction of this phenomenon was made by Kepler, - 
and was that of a transit of Mercury which occurred on the 7th 
of November, 1631. As early as 1629, Kepler announced to as- 
tronomers that his tables gave the latitude of Mercury, at the 
conjunction which was to take place on that day, less than the 
sun's semi-diameter ; consequently, that the planet in passing by 
the sun would be nearer the sun's center than the length of the 
sun's radius, and of course appear on his dis.k. The event cor- 
responded to the prediction. The latest transit of Mercury 
occurred on the 8th of November, 1848, being the 25th since the 
one predicted by Kepler, averaging nearly one in 8 years, although 
they take place at very unequal intervals. 

316. The shortest interval between two successive transits of 
Mercury is 3^ years, and of Venus 8 years ; but sometimes they 
are separated by long intervals, especially those of Venus. Not 
a single one of these will occur during the 20th century. The 
next transit of Mercury will take place November llth, 1861, 
and of Venus, December 8th, 1874. At the same node the 
shortest period for Mercury is 7 years ; but as there are two 
nodes, a transit may occur at one node 3j years after it oc- 
curred at the other. Thus there will be transits of Mercury in 
May, 1891, and November, 1894. More of the transits of Mer- 
cury happen in November than in May, because the orbit of 
this planet, (which has a great eccentricity, Art. 308,) is so sit- 
uated, that in November the planet is near its perihelion, and is 
then more likely to be projected on the sun in passing its inferior 
conjunction, than in a part of its orbit more distant from the 
sun. 

Let us see how the intervals between the transits of Mercurv 

* Hind. 



188 THE PLANETS. 

or Venus are found. Since Venus, for example, completes one 
revolution around the sun in 224.7 days, and the earth in 365.256, 
and since the number of times -each will revolve in a given pe- 
riod is inversely as the time of one revolution, therefore in 224,700 
revolutions of the earth, and 365,256 revolutions of Venus, the 
two bodies would meet exactly at the same node as before. But 
224,700 : 365,256 : : 8 : 13 nearly ; so that transits of Venus are 
sometimes repeated at intervals of 8 years, and if the ratio of 8 
to 13 were exactly that of the two first terms of the proportion, we 
should have a transit of Venus every 8 years. The ratio of 227 to 
369 is still nearer that of those terms ; and hence a transit after 
227 years is still more probable ; but since there are two nodes 
the chance is doubled, so that a transit is highly probable after- 
an interval of H3J years. The latest transit of Venus was that 
of June, 1769, one having previously occurred 8 years before; 
and the next transit will take place in December, 1874, and the 
next after that in December, 1882. From June, 1769, to Decem- 
ber, 1882, is a period of 113^ years ; but it so happens that Venus 
and the Earth will meet near enough to the node 8 years before 
to occasion a transit, thus anticipating the regular interval of 
1 13j years, and reducing it to 105^ years. If at the occurrence of 
a previous transit Venus had passed her node, the next transit, 
at the other node, happens 8 years sooner than the usual period 
of 113J years. f 

317. The great interest attached by astronomers to a transit 
of Venus, arises from its furnishing the most accurate means in 
our power of determining the suns horizontal parallax an ele- 
ment of great importance, since it leads to a knowledge of the 
distance of the earth from the sun, and consequently, by the ap- 
plication of Kepler's third law, (Art. 483,) of the distances of all 
the other planets. Hence, in 1769, great efforts were made 
throughout the civilized world, under the patronage of different 
governments, to observe this phenomenon under circumstances 
the most favDrable for determining the parallax of the sun. The 
method of finding the parallax of a heavenly body, described in 
Art. 85, cannot be relied on to a greater degree of accuracy than 



TRANSITS OF THE INFERIOR PLANETS. 189 

4". In the case of the moon, whose greatest parallax amounts to 
about 1, this deviation from absolute accuracy is not material; 
but it amounts to nearly half the entire parallax of the sun ; and 
since the distance is inversely as the horizontal parallax, such an 
error would make the distance of the sun either twice too great 
or twice too small, according as the parallax was 4" below or 4" 
above the truth. 



318. If the sun and Venus were equally distant from us, they 
would be equally affected by parallax as viewed by spectators in 
different parts of the earth, and consequently their relative situa- 
tion would not be altered by such a difference in the points of 
view ; but since Venus at the inferior conjunction is only about 
one-third as far off* as the sun, her parallax is proportionally 
greater, and therefore spectators, at distant points, will see Venus 
projected on different parts of the solar disk ; and as the planet 
traverses the disk, she will appear to describe chords of different 
lengths, by means of which the duration of the transit may be 
estimated at different places. The difference in -the duration of 
the transit, as viewed from opposite parts of the earth, does not 
amount to many minutes ; but to make it as large as possible, 
places very distant from each other are selected for observation. 
Thus, in the transit of 1769, among the places selected, two of 
the most favorable were Wardhus in Lapland, and Otaheite, (now 
written Tahiti,) one of the Society Islands, in the South Pacific 
Ocean, to which place the celebrated Captain Cook was dis- 
patched by the British government for the express purpose of 
observing the transit. 

Although the exact determination of the sun's horizontal par- 
allax by this method is a very complicated and difficult problem, 
yet the principle on which the process depends, admits of an 
easy illustration. Let E, (Fig. 63,) be the earth, V Venus, and 
S the sun. Suppose A and B two spectators at opposite extrem- 
ities of that diameter of the earth which is perpendicular to the 
ecliptic. The spectator at A will see Venus on the sun's disk at 
a, and ttie spectator at B will see Venus at b ; and since AV and 
BV may be considered as equal to each other, as also V6 and 



190 THE PLANETS. 

V; therefore the triangles are equiangular and similar, and 
AV : aV : : AB : db But the ratio of AV to aV is known, (Art. 
308 ;) hence the ratio of AB to ab is known, and when the an- 
gular value of ab as seen from the earth is found, that of AB 
becomes known as seen from the sun ; and half AB, or the semi 
diameter of the earth as seen from the sun, is the sun's horizon- 
Fig. 63. 




tal parallax, (Art. 82.) If, for example, ab is found to be 2^ 
times the diameter of the earth AB, or 5 times the semi-diameter, 
then, if the line AB be supposed to be on the sun, (for the sake 
of comparing it with ab,) it would subtend an angle at the eye 
equal to ^ of ab. But if viewed from the sun, the distance being 
the same, its apparent diameter would be the same, and ab would 
be five times the angular value of the semi-diameter of the earth 
as seen from the sun, and consequently (Art. 82) five times the 
sun's horizontal parallax. We have only then to find the angu- 
lar value of the line ab. We can ascertain the angular value of 
each chord EF or GH by the time occupied in describing it, since 
the motions of Venus and those of the sun are accurately known 
from the tables. Each chord being double the sine of half the 
arc cut off by it, therefore the sine of half the arc, and of course 
the versed sine becomes known, and the difference of the two 
versed sines ce (equal to cded)=ab. 

The appearance of Venus on the sun's disk being that of a 
well-defined black spot, and the exactness with which the mo- 
ment of external or internal contact may be determined, are cir- 
cumstances favorable to the accuracy of the result ; and astron- 
omers repose so much confidence in the estimation of the sun's 
horizontal parallax, as derived from the observations on the tran- 
sit of 1769, that this important element is thought to be certainty 



TRANSITS OF THE INFERIOR PLANETS. 191 

ascertained within one-tenth of a second. The general result of 
all these observations, gives the sun's horizontal parallax 8".6, or 
more exactly 8". 5776.* 

Venus when on the side of her inferior conjunction, and Mars 
when near his opposition, each comes comparatively near to the 
earth, and at these times exhibits a large horizontal parallax. 
That of Venus, especially, may be obtained with great accuracy 
when she is near her greatest elongation ; and since it is easy, 
by Article 308, to determine, at that time, the ratio of her dis- 
tance from the sun to the earth's distance, it is a matter of great 
interest to astronomy to have the parallax of Venus, when thus 
situated, accurately found. For this purpose, the government of 
the United States, in 1849, sent an expedition, under Lieutenant 
Gilliss, to Chili, in order to take observations on Mars and Venus, 
especially the latter, during 1850, 1851, and 1852, in concert with 
the Observatory at Washington. These researches, when com- 
pleted, will, it is hoped, afford a more accurate determination of 
the sun's horizontal parallax than any yet obtained. 

319. During the transits of Venus over the sun's disk in 1761 
and 1769, a sort of penumbral light was -observed around the 
planet by several astronomers, which was thought to indicate an 
atmosphere. This appearance was particularly observable while 
the planet was coming on and going off the solar disk. The 
total immersion and emersion were not instantaneous ; but as 
two drops of water when about to separate form a ligament 
between them, so there was a dark shade stretched out between 
Venus and the sun, and when the ligament broke, the planet 
seemed to have got about an eighth part of her diameter from the 
limb of the sun.f The presence of an atmosphere is also indica- 
ted by appearances of twilight and indications of a horizontal 
refraction. J 

Although no satellite has hitherto been discovered attending 
either Mercury or Venus, yet suspicions have, at different tinjes, 



* Delambre, t. 2. Vince, Complete Syst. vol. I. "Woodhouse, p. 754. Hersdhel'a 
Outlines, p. 255. 

f Ed. Encyc. Art. Astronomy. \ Hind. 



192 THE PLANETS. 

been entertained of a satellite belonging to Venus. None has 
been seen in any of the transits of Venus ; and although the. dis- 
tance of the satellite (if one exists) from the primary might have 
been too great to be projected with the primary on the sun, yet 
its absence on each of these occasions has strengthened the be- 
lief of astronomers that no such satellite exists. 



CHAPTER X. 

OF THE SUPERIOR PLANETS, MARS, JUPITER, SATURN, URANUS, AND 
NEPTUNE J AND OP THE NEW PLANETS, OR ASTEROIDS. 

320. THE Superior planets are distinguished from the Inferior, 
by being seen at all distances from the sun from to 180. 
Having their orbits exterior to that of the earth, they of course 
never come between us and the sun, that is, they have never 
any inferior conjunction like Mercury and Venus, but they 
are seen in superior conjunction and in opposition. Nor do 
they, like the inferior planets, exhibit to the telescope different 
phases, but, with a single exception, they always present the side 
that is turned towards the earth fully enlightened. This is owing 
to their great distance from the earth ; for were the spectator to 
stand upon the sun, he would of course always have the illumin- 
ated side of each of the planets turned towards him ; but so dis- 
tant are all the superior planets except Mars, that they are viewed 
by us very nearly as they would be if we actually stood on 
the sun. 



21. MARS is a small planet, his diameter being only about 

1 half that of the earth, or 4500 miles.* He also, at times, come? 

nearer to us than any other planet except Venus. His mean dis- 

tance is 145,200,000 miles ; but in consequence of the eccentricity 

of his orbit, the distance varies greatly, the difference between 

* Hind. 



MARS. 103 

the perihelion and aphelion distances being 27,000,000 miles. 
Mars is always near the ecliptic, never varying from it 2. He 
is distinguished from all the other planets by his deep red color 
and fiery aspect ; but his brightness and apparent magnitude vary 
much at different times', being sometimes nearer to us than at 
others by the whole diameter of the earth's orbit, that is, by 
about 190,000,000 miles. When Mars is on the same side of the 
sun with the earth, or at his opposition, he comes within 
50,000,000 miles of the earth, and, rising about the time the sun 
sets, surprises us by his magnitude and splendor ; but when he 
passes to the other side of the sun to his superior conjunction, he 
dwindles to the appearance of a small star, being then 240,000,000 
miles from us. Thus, let M, (Fig. 64,) represent Mars in opposi- 
tion, and M' in superior conjunction, it is obvious that the planet 

Fig. 64. 
M' 




must be nearer to us in the former situation than in the latter 
by the whole diameter of the earth's orbit. 

322. Mars is the only one of the superior planets which ex- 
hibits phases. When he is towards the quadratures at Q or Q/, 
it is evident from the fjgure that only a part of the circle of illu- 
mination is turned towards the earth, such a portion of the 
remoter part of it being concealed from our view as to render the 
form more or less gibbous. 

323. When viewed with a powerful telescope, the surface of 

25 



194 THE PLANETS. 

Mars appears diversified with numerous varieties of light and 
shade. The region around the poles is marked by white spots, 
which vary their appearance with the changes of the sea 
sons in the planet. Hence Dr. Herschel conjectured that they 
are owing to ice or snow which occasionally accumulates and 
melts, according to the position of each pole with respect to the 
sun.* It has been common to ascribe the ruddy light of this 
planet to an extensive and dense atmosphere, which was sup- 
posed to be distinctly indicated by the gradual diminution of 
light, observed in a star as it approached very near to the planei 
in undergoing an occultation ; but more recent observations afford 
no such evidence of an atmosphere. f By observations on the 
spots, we learn that Mars revolves on his axis in very nearly 
the same time with the earth, (24h. 39m. 21.3s.;) and that the 
inclination of his axis to the plane of his orbit, is also nearly the 
same, making his obliquity . 28 42', that of the earth being 23 
28', so that the changes of seasons in Mars must resemble 
our own. 

No satellite has ever been discovered belonging to Mars, 
although being situated at a greater distance from the sun 
than our globe, it might seem more especially to need such a 
luminary to cheer its dark nights. As the diurnal rotation of 
Mars is performed in nearly the same time as the earth, we 
should expect a similar flattening of j;he poles. Such is the 
fact, and the ellipticity of Mars exceeds that of the earth, being 
about one fiftieth,! while the earth's ellipticity is one three-hun- 
dredth. This difference in the conjugate diameters may be 
readily observed when the planet is in opposition, the whole 
enlightened disk being then presented to us. 

324. Mars being comparatively near to us when on the same 
side of the sun with the earth, and the ratio of his distance 
from the san to that of the earth being easily obtained, as- 
tronomers have sought by means of his parallax, as by that of 
Venus, to find the sun's horizontal parallax. But the method by 



* Phil. Trans. 1784. 

f Sir James South, PhiL Trans. 1833. t Hind 



JUPITER. 195 

observations on Venus, as described in Art. 318, is more to be 
relied on. 

325. JUPITER is distinguished from all the other planets by j 
his great magnitude. His diameter is 89,000 miles, being more 
than 11 times, and his volume more than 1400 times that of the 
earth. His figure is strikingly spheroidal, the equatorial exceed- 
ing the polar diameter in the ratio of 107 to 100,* which is 21 
times as great as the earth's ellipticity. This flattening of the 
poles is indeed quite perceptible by the telescope, and is obvious 
to the eye in a correct drawing of the planet. (See Frontis- 
piece.) Such a figure might naturally be expected from the 
rapidity of his diurnal revolution, which is accomplished in 
about 10 hours, (9h. 55m. 21.3s.)f 

A place on the equator of Jupiter must revolve 450 miles 
per- minute, or 27 times as fast as a place on the terrestrial equa- 
tor. The distance of Jupiter from the sun is 495,000,000 miles 
(495,8 17,000). J His axis of rotation is but slightly inclined to 
the plane of his orbit, (only about 3,) and consequently his cli- 
mate experiences but a slight change of seasons. 

326. The view of Jupiter through a good telescope, is one of 
the most magnificent and interesting spectacles among the heav- 
enly bodies. The disk expands into a large and bright orb like 
the full moon ; the spheroidal figure which theory assigns to re- 
volving worlds, is here palpably exhibited to the eye ; across the 
disk, arranged in parallel stripes, are discerned several dusky 
bands, called belts ; and four bright satellites, always in attend- 
ance, but ever varying their positions, compose a splendid retinue. 
Indeed, astronomers gaze with peculiar interest on Jupiter and 
his moons, as affording a miniature representation of the whole 
solar system, repeating, on a smaller scale, the same revolutions, 
and exemplifying, in a manner more within the compass of our 
observation, the same laws as regulate the entire assemblage of 
sun and planets. 

* HerscheL f Airy. $ Hind 



196 THE PLANETS. 

327. The Belts of Jupiter are variable in their number and 
dimensions. With smaller telescopes only one or two are seen 
across the equatorial regions ; but with more powerful instru- 
ments the number is increased, covering a great part of the disk. 
Occasionally these belts retain nearly the same form and posi- 
tions for many months together, while at other times they 
undergo great and sudden changes, and in one or two instances, 
they have been observed to break up and spread themselves over 
the whole face of the planet. The prevailing opinion among 
astronomers in reference to the nature of these belts is, that they 
are produced by disturbances in the planet's atmosphere, which 
occasionally render its dark body visible ; and, as the belts are 
found to traverse the disk in lines uniformly parallel to Jupiter's 
equator, they are inferred to be connected with the rotation of 
the planet upon its axis, the great rapidity of which would nat- 
urally produce peculiarities in its atmospheric phenomena. 

328. The Satellites of Jupiter may be seen with a telescope of 
very moderate powers. Even a common spy-glass will enable 
us to discern them. Indeed, being nearly equal in brilliancy to 
the smallest stars visible to the naked eye, a slight increase of 
optical power brings them into view ; and some few persons, en- 
dowed with extraordinary powers of vision, have supposed that 
they saw one of these little bodies without the aid of instru- 
ments ; but on applying the telescope it has been found that 
three of the satellites have approached so near together as to 
appear like one.* In the largest telescopes, they severally ap- 
pear as bright as Sirius does to the naked eye. With such 
an instrument, the view of Jupiter- with his moons and belts is 
truly a magnificent spectacle a world within itself. As the 
orbits of the satellites do not deviate far from the plane of the 
ecliptic, and but little from the equator of the planet, (which 
nearly coincides with the ecliptic,) they are usually seen almost 

* Hind. 

Rev. Mr. Stoddard, a graduate of Yale College, missionary to the Nestorians, 
has repeatedly seen one of these bodies with the naked eye, from Mount Seir, near 
Oroomiah. Mr. Stoddard is known to the author as an excellent observer, and his 
testimony on this point may be fully relied on. 



JUPITER. 



197 



in a straight line extending across the central part of the disk. 
(See Frontispiece.) 

329. Jupiter's satellites are distinguished from one another by 
the denominations of first, second, third, and fourth, according to 
their relative distances from the primary, the first being that 
which is nearest to him.* Their apparent motion is oscillatory, 
like that of a pendulum, going alternately from their greatest 
elongation on one side to their greatest elongation on the other, 
sometimes in a straight line, and sometimes in an elliptical 
curve, according to the different points of view in which we 
observe them from the earth. Their motion is alternately direct 
and retrograde ; they are sometimes stationary ; and, in short, 
they exhibit in miniature all the phenomena of the planetary 
system. Various particulars of the system are exhibited in the 
following table, the diameters being in miles, and the distances 
being taken from the center of the primary.f 



Satellite. 

1 

2 


Diameter. 


Distancea. 


Sidereal Revolution. 


2^:40 

2190 


278,500 
443,300 


Id. 18h. 28m. 
3 13 15 


3 


3580 


707,000 


7 3 43 


4 


3060 


1,243,500 


16 16 32 



Hence it appears, first, that Jupiter's satellites are all some- 
what larger than the moon, except the second, which is nearly 
of the same size with the moon. The third, the largest of the 
whole, has still only g^th the diameter of the primary. The 
greater distances also of these moons compared with ours, reduces 
their apparent size and light as seen from Jupiter. Thus the 
largest of them would exhibit to a spectator on the equator of 
the planet, a diameter of only 36', which is only a little greater 
than that of the moon, while the smallest would appear only one- 
fourth as large ; secondly, that the distance of the innermost 



* Mythological names were long since proposed for the satellites of Jupiter, 
viz., lo, Europa, Ganymede, Calisto : but the mode of designating them by numbers 
generally prevaila f Hind. 



198 THE PLANETS. 

satellite from the planet is but a little more than three times 
his diameter ; or if reckoned from the surface of the primary, 
nearly the same as the distance of the moon from the earth, 
while that of the outermost satellite is more than four times as 
far ; thirdly, that the first satellite completes its revolution around 
the primary in about a day and three quarters, while the fourth 
requires nearly sixteen days and th'ree quarters. 

330. The orbits of the satellites are nearly or quite circular, 
and deviate but little from the plane of the planet's equator, and 
of course are but slightly inclined to the plane of his orbit. They 
are, therefore, in a similar situation with respect to Jupiter as 
the moon would be with respect to the earth, if her orbit nearly 
coincided with the ecliptic, in which case she would eclipse the 
sun every new moon, and be herself eclipsed every full moon. 

331. The eclipses of Jupiter's satellites, in their general con- 
ception, are perfectly analogous to those of the moon, but in 
their details they differ in several particulars. Owing to the 
much greater distance of Jupiter from the sun, and its greater 
magnitude, the cone of its shadow is more than sixty times 
that of the earth, stretching off into space more than 55,000,000 
miles. On this account, as well as on account of the little incli- 
nation of their orbits to that of their primary, the three inner 
satellites of Jupiter pass through the shadow and are totally 
eclipsed at every revolution. The fourth satellite, owing to the 
greater inclination of its orbit, sometimes though rarely escapes 
eclipse* and sometimes merely grazes the limits of the shadow, 
or suffers a partial eclipse.* These eclipses, moreover, are not 
seen by us, as is the case with those of the moon, from the center 
of their motion, but from a remote station, and one whose situ- 
ation, with respect to the line of the shadow, is variable. This 
of course makes no difference in the times of the eclipses, but a 
very great difference in their visibility, and in their apparent sit- 
uations with respect to the planet at the moment of their enter- 
ing or quitting the shadow. 

* HerschersAstp.286 



JUPITER. 



199 



332. The eclipses of Jupiter's satellites present some curious 
phenomena, which will be best understood from a diagram. Let 
A, B, C, (Fig. 65,) represent the earth in different parts of its 
orbit, A being the western and C the eastern side, and B the place 
'of opposition. Let J represent Jupiter, a the first and b the fourth 
satellite ; and let xy represent the concave sphere of the heav- 
ens. When the earth is westward of the place of opposition, as 
at A, the immersions only are seen, the emersions being hidden 
behind the planet, as .will be evident on observing the rela- 



Fig. 65. 




tion of the satellite in passing through the shadow to the 
lines of vision drawn from the spectator to the primary and sec- 
ondary respectively. When the earth is eastward of the place 
of opposition, the emersions only are seen, as is also evident by 
conceiving lines drawn as before. This, however, is strictly true 
only of the first satellite ; for the third and fourth, and sometimes 
even the second, occasionally disappear and reappear on the 
same side of the disk. Thus, lines drawn from the eye to xy 
through b the place of immersion, and b 1 that of emersion, will 
strike the concave sphere of the heavens at c and d, while the 
planet will be seen at e. The same mode of illustration will show 
that when the earth is to the eastward of the planet, the immer- 
sions and emersions of the outermost satellite will be seen on the 
east side of the disk. When the earth is at B, the place of supe- 
rior conjunction, or at D, the place of opposition, -both the im- 



200 THE PLANETS. 

mersions and emersions take place behind the planet, and close 
to the disk. 

333. When one of the satellites is passing between Jupiter and 
the sun, it casts its shadow upon its primary, as the moon does 
on the earth in a solar eclipse, which is seen by the telescope 
travelling across the disk of Jupiter, as the shadow of the moon 
would be seen to traverse the earth by a spectator favorably sit- 
uated in space. When the earth is to the westward of Jupiter, 
as at A, the shadow reaches the disk of the planet, or is seen on 
the disk, before the satellite itself reaches it. Thus, (Fig. 65,) it 
will be seen that the line of projection drawn from A to any part of 
the shadow of the satellite, meets the planet sooner than the line 
drawn through the satellite ; and that just the opposite is the case 
when the earth has passed to C. We do not usually see the 
satellite itself projected on the disk of the primary, for, being 
illuminated like the primary, it is not readily distinguishable from 
it ; but sometimes, when it happens to be projected on one of 
the belts, it is seen, as a bright spot, making its transit across 
the disk. Occasionally, also, it is seen as a dark spot of smaller 
dimensions than the shadow. This curious fact has led to the 
conclusion that certain of the satellites have sometimes on their 
own bodies, or in their atmospheres, obscure spots of great 
extent.* 

334. A very singular relation subsists between the mean mo- 
tions of the three first satellites of Jupiter. The mean longitude 
of the first, plus twice that of the third, minus three times that 
of the second, always equals 180 degrees. A curious consequence 
of this relation is, that the three satellites can never be all 
eclipsed at the same time ; for then, having severally the same 
longitude as the primary, their longitudes would be equal, and 
that of the first, plus twice that of the third, minus three times 
that of the second, would be nothing, and of course could not be 
180 degrees. f These phenomena are such as would present 
themselves to a spectator on Jupiter, and not to a spectator on 
the earth. 

* Sir J. HerscheL f Biot 



JUPITER. 201 

335. The discovery of the system of Jupiter and his satellites, 
soon after the invention of the telescope, lent a powerful support 
to the Copernican system of astronomy, then just beginning to be 
received by astronomers, since it presented to the eye an exact 
miniature of the solar system, and exhibited an actual model of 
that arrangement of the sun and planets which had before only 
been contemplated by the eye of the mind ; and the laws of the 
planetary system, discovered by Kepler, were here actually seen 
to be verified, in the. motions of this miniature system. More 
over, the eclipses of Jupiter's satellites, furnished one of those in- 
stantaneous events, occurring at the same moment of absolute 
time wherever seen, which are available for finding the longitudes 
of different places ; and at that period, it was deemed a more 
eligible method of determining this great practical problem of 
astronomy, than any method then in use. 

336. The eclipses of these satellites seem to have various 
requisites for determining longitudes, being, as already remarked, 
seen at the same moment at all places where the planet is vis- 
ible, being wholly independent of parallax, and being predicted 
beforehand with great accuracy the instant they occur at Green- 
wich, and given in the Nautical Almanac : but several circum- 
stances conspire to render tHis method of finding the longitude 
less eligible than several other methods at present in use. The 
extinction of light in the satellite at its immersion, and the re- 
covery of its light at its emersion, are not instantaneous but 
gradual ; for the satellite, like the moon, occupies some time in 
entering into the shadow or in emerging from it ; which occasions 
a progressive diminution or increase of light. The better the 
light afforded by the telescope with which the observation is 
made, the later the satellite will be seen at its immersion, and the 
sooner at its emersion.* In noting the eclipses even of the first 
satellite, the time must be considered as uncertain to the amount 
of 20 or 30 seconds ; and those of the other satellites involve 
still greater uncertainty. Two observers, in the same room, ob- 



* This is the reason -why observers are directed, in the Nautical Almanac, to 
telescopes of a certain power. 

26 



202 THE PLANETS. 

serving, with different telescopes, the same eclipse, will frequently 
disagree in noting its time to the amount of 15 or 20 seconds, and 
the difference will always be the same way.* Better methods, 
therefore, of finding the longitude are now employed, although 
the facility with which the necessary observations can be made, 
and the little calculation required, still render this method eligible 
in many cases where extreme accuracy is not required. As a 
telescope is essential for observing an eclipse of one of these sat- 
ellites, this method cannot be practised at sea. 

337. The grand discovery of the progressive motion of light, 
I was first made by observations on the eclipses of Jupiter's satel- 
lites. In the year 1675, it was remarked by Roemer, a Danish 
astronomer, on comparing together observations of these eclipses 
during many successive years, that they take place sooner by 
about sixteen minutes (10m. 26.6s.) when the earth is on the 
same side of the sun with the planet, than when she is on 
the opposite side. This difference he ascribed to the progres- 
sive motion of light, which takes that time to pass through the 
diameter of the earth's orbit. Now, 16m. 26.6s. = 986.6s. 
/. 986.6 sec. : 190,000,000 miles : : 1 sec. : 192,600 miles = the 
velocity of light per second, equal to nearly 12,000,000 miles per 
minute. So great a velocity startled astronomers at first, and 
produced some degree of distrust of this explanation of the phe- 
nomenon ; but the subsequent discovery of the aberration of light, 
(Art. 195,) led to an independent estimation of the velocity of 
light with nearly the same result. 

338. SATURN comes next in the series as we recede from the 
sun, and has, like Jupiter, a system within itself, on a scale of 
great magnificence. In size it is, next to Jupiter, the largest of 
the planets, being 79,000 miles in diameter, or nearly 10 times 
as large as the earth in diameter, and about 1000 times as large 
in volume. It has likewise belts on its surface, and is attended 
by eight satellites. But a still more wonderful appendage is its 
Ring, a broad wheel encompassing the planet at a great distance 

* "Woodhouse, p. 840. 



SATURN. 203 

from it. We have already intimated that Saturn's system is on 
a grand scale. As, however, Saturn is distant from us nearly 
900,000,000 miles, we are unable to .obtain the same clear and 
striking views of his phenomena that we do of the phenomena of 
Jupiter, although they really present a more wonderful mechan- 
ism. The figure of Saturn has usually been described, on the 
authority of Sir William Herschel, as approaching that of a cube ; 
but more recent and refined measurements have shown that it is 
elliptical, being much compressed at the poles, the equatorial 
exceeding the polar diameter by about one-tenth.* 

The belts of Saturn, although clearly discerned by a good tel- 
escope, are far more indistinct than those of Jupiter. Spots, 
which occasionally appear on the belts, have enabled astronomers 
to determine the time of the diurnal rotation of Saturn, which is 
found to be about ten hours and a half, (lOh. 29m.) 

> 

339. Saturn's ring, when viewed with powerful telescopes, is 
found to consist of two concentric rings, separated from each 
other by a dark space. f (See Frontispiece.) Although this 
division of the rings appears to us, on account of our immense 
distance, as only a fine line, yet it is in reality an interval of nearly 
1800 miles. The dimensions of the whole system are in round 
numbers, as follows :J 

Miles. 

Diameter of the planet, .... 79,000 

From the surface of the planet to the inner ring, 20,000 
Breadth of the inner ring, . . . 17,000 

Interval between the rings, . . . 1,800 

Breadth of the outer ring, . . . 10,500 

Extreme dimensions from outside to outside 176,000 

The figure represents Saturn as it appears to a powerful teles- 
cope, surrounded by its rings, and having its body striped with 

* Hind. 

j- A greater number of divisions of the rings have been occasionally seen, and the 
researches of Mr. GK P. Bond and of Professor Peirce,' render it probable that the 
number is variable. (See Trans. Amer. Academy.) 

t Professor Struve, Mem. Art. Soc. III. 301. 



204 THE PLANETS. 

i 

dark belts, somewhat similar, but broader and less strongly 
marked, than those of Jupiter, and owing doubtless to a similar 
cause.* That the ring is composed of matter of considerable 
density, is shown by its throwing a deep shadow on the body of 
the planet on the side nearest the sun, and on the other side re- 
ceiving that of the body.f 

From the parallelism of the belts with the plane of the ring, it 
may be conjectured that the axis of rotation of the planet is per- 
pendicular to that plane; and this conjecture is confirmed by 
the occasional appearance of extensive dusky spots on its surface, 
which, when watched, indicate a rotation parallel to the ring in 
about' ten hours and a half, (lOh. 29m. 17s.) This motion, it will 
be remarked, is nearly the same with the diurnal motion of Jupi- 
ter, subjecting places on the equator of the planet to a very swift 
revolution, and occasioning its striking spheroidal figure ; and 
the axis of rotation, like that of the earth, preserves its parallelism 
to itself during the motion of the planet in its orbit. According 
to Sir William Herschel, the planet is surrounded with a very 
dense atmosphere, which is indicated by the refraction experi- 
enced by the satellites when they are passing behind the planet, 
and by periodical changes of color and shade in the polar 
regions. 

It requires a telescope of high magnifying powers and a strong 
light to give a full and striking view of Saturn with his rings and 
belts and satellites ; for we must bear in mind that at that dis- 
tance, one second of angular measurement corresponds to 4000 
miles, a space equal to the semi-diameter of our globe. But with 
a telescope of moderate powers, the leading phenomena of the 
ring itself may be observed. 

340. Saturn's ring, in its revolution around the sun, always 
remains parallel to itself. 

If we hold opposite to the eye a circular ring or disk, like a 



* Sir J. Herschel. 

f Recent investigations of Mr. George P. Bond, of the Observatory of Harvard 
University, and of Professor Peirce, indicate that the rings are composed of matter 
in the fluid state. 



SATURN. 205 

piece of coin, it will appear as a complete circle when it is at 
right angles to the axis of vision ; but when oblique to that axis, 
it will be projected into an ellipse more and more acute as its 
obliquity is increased, until, when its plane coincides with the 
axis of vision, it is projected into a straight line. Let us place 
on the table a lamp or a ball to represent the sun, and, holding 
the ring at a certain distance, inclined a little towards the cen- 
tral body, let us carry it round, always keeping it parallel to 
itself. During its revolution it will twice present its edge to the 
lamp or ball at opposite points, and twice at 90 distance from 
those points, it will present its broadest face towards the central 
body. At intermediate points, it will exhibit an ellipse more or 
less open, according as it is nearer one or the other of the pre- 
ceding positions. It will be seen also that in one half of the 
revolution, the lamp shines on one side of the ring, and in the 
other half of the revolution on the other side. Such would be 
the successive appearances of Saturn's ring to a spectator on the 
sun ; and since the earth is in respect to so distant a body as 
Saturn, very near the sun, those appearances are presented to 
us nearly in the same manner as though we viewed them from 
the sun. Accordingly, we sometimes see Saturn's ring under the 
form of a broad ellipse, which grows continually more and more 
acute until it passes into a line, and we either lose sight of it alto- 
gether, or, with the aid of the most powerful telescopes, we see 
it as a fine line drawn across the disk, and projecting out from it 
on each side. As the whole revolution occupies nearly 30 years, 
and the edge is presented to us twice in the revolution, this last 
phenomenon, namely, the disappearance of the ring, takes place 
every 15 years, when sometimes two and sometimes three dis- 
appearances occur very near together. 

341. The learner may perhaps gain a clearer idea of the fore- 
going appearances from the following diagram. 

Let A, B, C, &c., represent successive positions of Saturn and 
his ring in different parts of his orbit, while ab represents the 
orbit of the earth.* Were the ring when at C and G perpendic- 

* It may be remarked by the learner, that these orbits are made so elliptical, not 



206 THE PLANETS. 

ular to the line joining CG, it would be seen by a spectator situated 
at a or b as a perfect circle, but being inclined to the line of vision 
28 II 7 , it is projected into an ellipse. This ellipse contracts in 
breadth as the ring passes towards its nodes at A and E, where 
it dwindles into a straight line. From E to G the ring opens 

Fig. 66. 




again, becomes broadest at G, and again contracts until it be- 
comes a straight line at A, and from this point it expands until it 
recovers its original breadth at C, in which case the breadth is 
very nearly half the length of the ellipse. These successive ap- 
pearances are all visible in a telescope of moderate powers, as 
represented in the foregoing diagram. 

342. The several circumstances which occasion the disappear 
ance of the ring two or three times within a short period everj 
fifteen years, may be understood from the following explanation 
Let S, (Fig. 67,) be the sun, ABCD a part of Saturn's orbit 
which includes the node at C, and CS will, of course, be the line 
of the nodes. Let EFGH be the earth's orbit, and EB, GD, lines 
parallel to CS, and touching the earth's orbit in E and G. Since 
the ring always remains parallel to itself, its plane can nowhere 
present its edge to the earth's orbit, except when the planet is 
between B and D, during which time and then only can a disap- 
pearance take place. Since Saturn is 9.54 times as far from the 
sun as the earth is, therefore, 



to represent the eccentricity of cither the earth's or Saturn's orbit, but merely 
the projection of circles seen very obliquely. 



SATURN. 



207 



9.54 : 1 : : Rad : Sin. SBE^G 1'^BSC, and the whole angle 
BSD=12 2', an arc which is described by Saturn at his mean 
rate, in 359.46 days, or nearly a year, of which it falls short only 
about 5f days. Let the earth set out from G when the planet 
sets out from B, and let Ga be the arc of the earth's orbit de- 
scribed from G in 5f days. Then, if at the moment of Saturn's 



Fig. 67. 




arrival at B, the earth is at a, a spectator on the earth will see 
the plane of the ring advancing parallel to itself towards him, and 
will come into such a position that its edge will be presented to 
him somewhere in the quadrant HE, since the earth will de- 
scribe half its entire orbit while the ring is moving from B to C. 
Let M be the point where the earth passes the ring. It will then 
be on the dark side of the ring, and continue so until the ring 
has passed the sun at C, when it will again become visible, and 
remain visible until the earth again comes up with it at G. In 
this case there will be two disappearances, one while the ring is 
moving from K to C, a period of considerable duration, (the dark 
side being all this while turned towards the earth,) and the other 
but momentary, since the earth overtakes it just at the moment 
of the planet's quitting the arc BD, beyond which its edge can 
nowhere be presented towards the earth's orbit. If, when Sat- 
urn is at B, the earth is in any part of the arc HE, it will 
meet and pass the ring in the quadrant HE ; and the earth will 
overtake it before it reaches D, and passing round G, will meet it 
again in the quadrant GH ; so that in this case there will be 
three disappearances of the ring in the course of a year. But 
should the earth be at E when the ring is at B, the motion of the 
earth being at that time directly towards the ring, the latter will 



208 THE PLA1TETS. 

leave it behind, (still presenting its dark side to the earth ;) but 
the earth, by its more rapid revolution, will soon pass the ring 
somewhere in the quadrant EF, and again in the quadrant GH. 
But before the earth has made another entire circuit, the ring 
will have advanced beyond D, so that, in this case, there will 
be two disappearances. It appears, therefore, that there are 
three causes for the disappearance of Saturn's ring ; first, when 
the edge of the ring is presented to the sun ; secondly, when the 
edge is presented to the earth ; and, thirdly, when the unillumi- 
nated side is towards the earth. 

343. Saturn's ring revolves in its own plane in about 10 \ 
hours, (lOh. 32m. 15.4s.) La Place inferred such a revolution 
from the doctrine of universal gravitation. He proved that such 
a rotation was necessary, otherwise the matter of which the ring 
is composed would be precipitated upon its primary. He showed 
that in order to sustain itself, its period of rotation must be equal 
to the time of the revolution of a satellite, circulating around 
Saturn at a distance from it equal to that of the middle of the 
ring, which period would be about 10J hours. By means of spots 
on the ring, Dr. Herschel followed it in its rotation, and actually 
found its period to be the same as assigned by La Place.* 

The thickness of the ring, according to Sir John Herschel, 
does not exceed a hundred miles. It is not quite concentric with 
the body of the planet, an arrangement which is essential to its 
stability, since, were it perfectly circular, of uniform density, and 
concentric with the planet, it would be in a condition of unstable 
equilibrium, ready to fall on the planet by the least disturbing 
force, like the attraction of one of the satellites, f 

Within the double ring of Saturn, as exhibited to ordinary tel- 
escopes, there has recently been discovered a new ring, less 
luminous than the others, and therefore concealed from previous 
observers. This was first discovered by Mr. G. P. Bond, with 
the great refractor of Harvard Observatory. 

344. The rings of Saturn must present a magnificent spectacle 
* System du Monde, 1. iv. c. 8. f Herschel's Outlines, p. 279. 



SATURN. 209 

from those regions of the planet which lie on their illuminated 
sides, appearing as vast arches spanning the sky from horizon to 
horizon, and holding an invariable, situation among the stars. 
On the other hand, in the region beneath the dark side, a solar 
eclipse, of fifteen years' duration, must afford an inhospitable 
abode to animated beings, but ill-compensated by the full light of 
its satellites.* 



345. Saturn is attended by eight satellites. Although bodies of 
considerable size, varying from 500 to 2850 miles,f their great 
distance prevents their being visible to any telescopes but such 
as afford a strong light and high magnifying powers. The outer- 
most satellite is distant sixty-four times the radius of the primary, 
a reach of 2,500,000 miles. The whole extent, therefore, of the 
system of Saturn is immense a realm within itself, being from 
side to side nearly five millions of miles. When represented in 
a diagram, on a scale in which the diameter of the planet is only 
one foot, the satellites reach out through the long line of thirty- 
two feet on each side. It is only representations of this kind 
that give any just ideas of the amplitude of the celestial system, 
while the contracted and crowded figures of ordinary diagrams, 
or even of orreries, help to form only erroneous and wholly inade- . 
quate views of these systems. 

, The names of the satellites of Saturn are Mimas, Ence- 
ladus, Tethys, Dione, Rhea, Titan, Hyperion, and Japetus. The 
seventh, Hyperion, was recently discovered by Professor Bond, 
with the great Cambridge refractor. At the time of the disap- 
pearance of the rings, (to ordinary telescopes,) the satellites were 
seen by Sir William Herschel, with his great telescope, projected 
along the edge of the ring, and threading, like beads, the thin 
fibre of light to which the ring is then reduced. Owing to the 
obliquity of the ring and of the orbits of the satellites to that of 
the primary, there are no eclipses of the satellites, the two interior 
ones excepted, until near the time when the ring is seen edge- 
wise.J 



* Sir J. Herschel. f Hind. \ Sir J. HerscheL 

27 



210 THE PLANETS. 

346. URANUS, the next planet in the series, was discovered by 
Sir William Herschel, in 1781. Previous to this time, Saturn 
had, from a high antiquity, been considered as the outermost boun- 
dary of the solar system ; but this discovery doubled the dimen- 
sions of the system, bringing to light a large planet at about twice 
the distance of Saturn from the sun, and about 19 times the dis- 
tance of the earth, or 1800 millions of miles. . It was named by 
the discoverer the Georgian, in honor of his patron George III. ; 
but this name being unacceptable to astronomers of other coun- 
tries, the planet was called Herschel in America, after the name 
of the discoverer, and Uranus* on the continent of Europe, which 
last appellation is now universally adopted. The diameter of 
Uranus is about 35,000 miles, and consequently its volume more 
than 80 times that of the earth. Its revolution around the sun 
occupies nearly 84 years, so that its position among the stars 
varies but little for several years in succession, since it shifts its 
place only a little more than four degrees in a year, and of course 
would remain in the same sign of the Zodiac seven years. Its 
path lies very near the ecliptic, being inclined to it less than 
47'. The sun himself, when seen from Uranus, dwindles al- 
most to a star, subtending, as it does, an angle of only 1' 40" ; so 
that the surface of the sun would appear there nearly 400 times 
less than it does to us. 

347. The satellites of Uranus are exceedingly minute objects, 
and visible only to the most powerful telescopes. Although Sir 
William Herschel assigned six satellites to this planet, yet only 
two of the number (the second and fourth in the order of dis- 
tances) have, until quite recently, been seen by other astrono- 
mers. Two others have of late been added, and an increasing 
confidence is beginning to be felt that the entire number an- 
nounced by Herschel will be identified. The orbits of these sat 
ellites, says Sir John Herschel, offer remarkable, and indeed quite 
unexpected and unexampled peculiarities. Contrary to the un- 
broken analogy of the whole planetary system, whether of pri- 
maries or secondaries, the planes of their orbits are nearly 

* From ovpavos, the father of Saturn. 



NEPTUNE. 211 

perpendicular to the ecliptic, being inclined no less than 78 58' 
to that plane, and in these orbits their motions are retrograde. 
Instead of advancing from west to east, as is the case with every 
other planet and satellite, they remove in the opposite direction, or 
from east to west. With this exception, all the motions of the 
planets, whether around their own axes or around the sun, and 
that of the sun himself on his axis, are from west to east. 

348. NEPTUNE is (so far as is known) the 'last planet of the series, j 
being removed from the sun to the immense distance of nearly 
3000 millions of miles (2,862,457,000). Its diameter is a little less 
than that of Uranus, being 31,000 miles.* Its volume is nearly 
sixty times that of the earth. Its periodic time is 164^ years, 
which is about twice that of Uranus. Its orbit is nearly circu 
lar, and but little inclined to the ecliptic, (1 47'.) 

The discovery of the planet Neptune is the most remarkable 
astronomical event of our times, and is generally considered as the 
most extraordinary discovery ever made in physical science. The 
leading steps of the process were as follows. The planet Ura- 
nus had long been known to be subject to certain irregularities 
in its revolution around the sun, not accounted for by all the 
known causes of perturbation. In some cases the deviation from 
the true place, as given by the tables, differs from actual obser- 
vation two minutes of a degree a quantity indeed which seems 
small, but which is still far greater than occurs in the case of 
the other planets, and far too great to satisfy the extreme ac- 
curacy required by modern astronomy. This fact long since 
suggested to astronomers the possibility of one or more addi- 
tional planets, hitherto undiscovered, which, by their attractions, 
exert on Uranus a great disturbing influence. Le Verrier, a dis- 
tinguished French astronomer, assuming the existence of such a 
planet, applied himself, by the aid of the calculus, guided by the 
law of universal gravitation, to the inquiry where the hidden 
planet was situated at what distance from the sun and at what 
point of the starry heavens ? From Bode's law of the planetary 
distances, (Art. 299,) according to which Saturn is nearly twice 

* Hind. 



212 THE PLANETS. 

as far from the sun as Jupiter, and Uranus twice as far as Sat- 
urn, he inferred that, if a planet exists beyond Uranus, its distance 
is probably about twice that of Uranus, or about 3600 millions of 
miles from the sun, which is nearly thirty-eight times that of 
the earth. He assumed it, however, to be thirty-six times the 
earth's mean distance. The corresponding periodic time would 
be 216 years. After reasoning from analogy, and the doctrine 
of universal gravitation, respecting the position and mass which 
a body must have in order to account for the perturbations of 
Uranus, equations were formed between these perturbations and 
the elements of the body in question, both known and unknown. 
These equations were exceedingly complex and difficult of reduc- 
tion ; but, by the most ingenious artifices, the several unknown 
quantities were successively eliminated, either directly or by re- 
peated approximations, until the great geometer arrived at 
expressions for the elements of the unknown planet, which indi- 
cated its place among the stars, its quantity of matter, the shape 
of its orbit, and the period of its revolution. Having placed the 
body in various positions in the orbit thus determined, he found 
that when situated at a point in the constellation Capricornus, 
its effect upon Uranus would be such as corresponded to the irreg- 
ularities to be accounted for ; that on the 1st of January, 1847, 
the hidden planet would have a longitude of 326 32', and vffculd 
lie about five degrees eastward of the well-known star Delta Ca- 
pricorni. He further asserted that it would have an apparent 
diameter of about 3", and therefore be visible to large telescopes. 

349. Having communicated these results to the French Acad- 
emy, at their sitting on the 31st of August, 1846, Le Verrier soon 
afterwards made them known to Dr. Galle, one of the astronomers 
of the Royal Observatory of Berlin, with the request that he would 
search for the stranger with the powerful telescope at his com- 
mand. On the same evening that Dr. Galle received the com- 
munication, namely, on the 23d of September, he directed his 
telescope towards the spot assigned for the planet, and there 
it was, within less than a degree of the place indicated by Le 
Verrier, and having an apparent magnitude within half a second 
of that assigned. To show the near correspondence between 



NEPTUNE. 213 

theory and observation, we may remark that the predicted lon- 
gitude, for the 23d of September, at midnight, was 324 58', and 
the observed longitude was 325 52'.8 ; the predicted diurnal 
motion in longitude was 69", and the observed 74". These re- 
sults struck the scientific world with astonishment, and their 
confirmation was one of the greatest achievements of the human 
mind. 

. . 

350. It has often happened, in the history of great discoveries, 
that the same hidden truth is revealed simultaneously to different 
inquirers, and accordingly, by a singular coincidence, a young 
mathematician of the University of Cambridge, (Eng.,) Mr. 
Adams, had, without the least knowledge of what M. Le Verrier 
was doing, arrived at the same great result. But having failed 
to publish his paper until the world was made acquainted with 
the facts through the other medium, he has lost much of the 
honor which the priority of discovery would have gained for 
him. Thus two distinguished mathematicians, unknown to each 
other, and by entirely independent processes, had arrived at the 
same results, as regarded both the existence of the supposed planet, 
and the region of the starry heavens where at that moment it 
lay concealed ; and, to crpwn all, astronomers, in obedience to 
the direction of one of them, had pointed their telescopes to the 
spot, and found it there. The conviction on the mind of every 
one was, that nothing but absolute truth could abide a test so 
unequivocal. It still remained, however, to determine by obser- 
vation whether the body actually conformed, in all respects, to 
the results of theory. To settle this point completely, that is, to 
determine with precision the elements of the orbit from observa- 
tion, would require a long time in a planetary body whose mo- 
tion was so slow that more than two centuries, as was supposed, 
would be required to complete a single revolution. But if it 
should be found that, among preceding catalogues of the stars, 
this body might have been included, and its place recorded as a 
fixed star, then, by comparing that place with its present posi- 
tion, and noting the interval of time between the two observa- 
tions, we might thus learn the rate of its motion, and its peri- 
odic time, and might thence deduce various other particulars 



214 THE PLANETS. 

dependent on these elements. Our distinguished countryman, 
Mr. Sears C. Walker, then connected with the observatory at 
Washington, undertook this investigation. First, from the ob- 
servations already accumulated, he calculated the path which 
the planet must have pursued for the last fifty or sixty years, and, 
by tracing this path among the stars of Lalande's catalogue, he 
found that it passed within two minutes of a star of the seventh 
magnitude, which was recorded as being seen in May, 1795. 
Professor Hubbard, of the same observatory, on reconnoitering 
for this star, found that it was missing. Little* doubt remained 
that the star seen by Lalande, was the planet of Le Verrier ; and 
this conclusion was confirmed by calculating its orbit on this 
supposition, and comparing the results with the places it has 
actually occupied since it fell within the sphere of observation. 
The results thus obtained, however, were materially different 
from those of Le Verrier and Adams. Instead of a period of 
216 years, they give only a period of 164^ years ; and instead 
of a distance of 3600 millions of miles, the new period would re- 
quire a distance of only 2862 millions. The eccentricity of the 
orbit, moreover, according to Walker, is much less than had been 
assigned to it, the orbit being in fact very nearly circular, while, 
by Le Verrier's estimate, it was considerably elliptical. The 
longitude, in fact, proved to be nearly the same as that assigned 
to it, and this rendered the discovery of it with the telescope 
so easy. The elements thus corrected account fully and com- 
pletely for the irregularities of Uranus sought to be explained, 
within a single second, as determined by Professor Peirce.* 



NEW PLANETS, OR ASTEROIDS* 

351. The commencement of the present century was rendered 
memorable in the annals of astronomy, by the discovery of four 
new planets between Mars and Jupiter. Kepler, from some 
analogy which he found to subsist among the distances of the- 
planets from the sun, had long before suspected the existence of 
a planet at this distance ; and his conjecture was rendered more 

* Amer. Journal of Science, New Series, voL v. p. 486. 



NEW PLANETS, OR ASTEROIDS. 215 

piobable by the discovery of Uranus, which follows the analogy 
of the other planets. So strongly were astronomers impressed 
with the idea that a planet would be found between Mars and 
Jupiter, that, in hope of discovering it, an association was formed 
on the continent of Europe of twenty-four observers, who divi- 
ded the sky into as many zones, one of which was allotted to 
each member of the association. The discovery of the first of 
these bodies was, however, made accidentally by Piazza, an as- 
tronomer of Palermo, on the 1st of January, 1801. It was 
shortly afterwards lost sight of, on account of its proximity to 
the sun, and was not seen again until the close of the year, when 
it was rediscovered in Germany. Piazza called i't Ceres, in 
honor of the tutelary goddess of Sicily, and her emblem, the 
sickle 9 , has been adopted as the appropriate symbol. The dif- 
ficulty of finding Ceres, induced Dr. Olbers, of Bremen, to 
examine, with particular care, all the small stars that lie near her 
path, as seen from the earth ; and, while prosecuting these obser- 
vations, in March, 1.802, he discovered another similar body, 
very nearly at the same distance from the sun, and resembling 
the former in many other particulars. The discoverer gave to 
this second planet the name of Pallas, choosing for its symbol 
the lance $ , the characteristic of Minerva. 

352. The most surprising circumstance connected with the 
discovery of Pallas, was the existence of two planets at nearly 
the same distance from the sun, and apparently having a common 
node ; a circumstance that indicated an identity of origin. On 
account of this singularity, Dr. Olbers was led to conjecture that 
Ceres and Pallas are only fragments of a larger planet which had 
formerly circulated around the sun at this distance, and been 
shattered by some great convulsion. 

In 1804, near one of the nodes of Ceres and Pallas, a third 
planet was discovered. This was named Juno, and the charac- 
ter o was adopted for its symbol, representing the starry sceptre 
of the goddess. In 1807, a fourth planet, Vesta, was discovered, 
and for its symbol the character fi was chosen an altar sur- 
mounted with a censer holding the sacred fire. It is the largest 
of the asteroids, and has sometimes been seen by the naked eye. 



216 THE PLANETS. 

353. From 1807 to 1845, a periocl of nearly forty years, no 
more of these small planets were discovered, and, up to this time, 
by the asteroids were meant the four little planets already enu- 
merated Ceres, Pallas, Juno, and Vesta. Meanwhile, very 
accurate maps of the stars, including all up to the tenth magni- 
tude, had been published, especially in the region of the zodiac, 
and astronomers scrutinized these with such extreme closeness 
that any wanderer appearing among them, was likely to be im- 
mediately detected. Since 1845 to the present time, (January, 
1855,) no fewer than 29 more asteroids have been discovered, 
making the entire number at present 33, as enumerated in arti- 
cle 296. The average distance of the asteroids from the sun, is 
about 2\ times that of the earth, or 240,000,000 miles ; but these 
distances vary considerably among themselves Flora being only 
about 200, and Hygeia nearly 300 millions of miles from the sun. 
As they are found to be governed by Kepler's law, like the other 
members of the solar system, their average time of revolution 
about the sun is nearly 4 years ; although the nearest asteroid 
completes its period in a little more than 3, while the most dis- 
tant requires about 5j years. Some of these bodies have their 
orbits much more eccentric and highly inclined to the ecliptic than 
those of the old planets. Juno and Pallas move in orbits more 
eccentric even than that of Mercury ; and the inclination of 
Vesta exceeds 34 degrees, while those of several others are much 
more highly inclined than the orbit of Mercury. Their small 
size constitutes one of their most remarkable peculiarities. The 
difficulty of estimating the apparent diameter of bodies at once 
so very small and so far, would lead us to expect that the esti- 
mates of different observers would vary ; but all agree that their 
diameters are only a few hundred miles at most. 

354. We have waited until the learner may be supposed to be 
familiar with the heavenly bodies, individually, before inviting his 
attention to a systematic view of the planets in their revolutions 
around the sun, and their grand laws. The time has now ar- 
rived for entering more advantageously upon this subject than 
could have been done at an earlier period. 

There are two methods of arriving at a knowledge of the MO- 



MOTIONS OF THE PLANETARY SYSTEM. 217 

TIONS of the heavenly bodies. One is, to begin with the apparent, 
and from these to deduce the real motions ; the other is to begin 
with considering things as they really are in nature, and then to 
inquire why they appear as they do. The latter of these methods 
is by far the more eligible. It is much easier than the other : 
and proceeding from the less difficult to that which is more so 
from motions which are very simple to such as are complicated, 
it finally puts the learner in possession of the whole machinery of 
the heavens. We shall in the first place, therefore, endeavor to 
introduce the student to an acquaintance with the simplest mo- 
tions of the planetary system, and afterwards to conduct him 
gradually through such as are more complicated and difficult. 

355. When viewed from the center of their motions, the revo- 
lutions of the planets would appear simple and harmonious, all 
coursing around the spectator from west to east in regular order, 
in nearly the same great highway, though with very different de- 
grees of velocity. Let us, then, suppose ourselves standing on 
the sun, and contemplate the revolutions of the planets, first, sev- 
erally, and then as forming one grand whole, consisting of nu- 
merous parts, but bound together under the same laws in one 
vast empire. We should see Mercury making very perceptible 
progress from night to night, like the moon in its motions about 
the earth, his daily progress eastward being about one-third as 
great as that of the moon, since he completes his entire revolu- 
tion in about three months. It will, at first, aid our conceptions 
of the respective positions of the planetary orbits, to imagine the 
ecliptic, to be marked out on the face of the visible heavens in a 
palpable line distinctly visible to the eye. If we watch the mo- 
tions of Mercury from night to night, we shall see it cross the 
ecliptic in two opposite points of the heavens, constituting its 
nodes ; and we shall see it, when half way between the nodes, at 
an angular distance from the ecliptic of about 7, this being the 
inclination of its orbit. Knowing the position of the orbit of 
Mercury with respect to the ecliptic, we may now, in imagina- 
tion, represent that orbit in a great circle passing through the 
centre of the planet and the center of the sun, and cutting the 
plane of the ecliptic in two opposite points in an angle of 7 de- 

28 



218 THE PLANETS. 

grees. The planes of both the ecliptic and the orbit of Mercury 
may be conceived of as indefinitely extended until they meet the 
sphere of the fixed stars ; but the lines which the earth and Mer- 
cury describe in those planes, that is, their orbits, may be con- 
ceived of as comparatively near to the sun. Could we now for 
a moment be permitted to imagine that the planes of the earth's 
orbit, and of the orbit of Mercury, were made of thin plates of 
glass, and that the paths of the respective planets -were marked 
out on their planes in distinct lines, we should perceive the orbit 
of the earth to be almost a perfect circle, while that of Mercury 
would appear distinctly elliptical, and we should see visibly rep- 
resented to the eye the several relations of these two orbits to 
each other. But having once made use of a palpable surface and 
visible lines to aid us in giving position and figure to the plane- 
tary orbits, let us now throw aside these devices, and hereafter 
conceive of these planes and orbits as they are in nature, and 
learn to refer a body to a mere mathematical plane, and to trace 
its path in that plane through absolute space. 

356. A clear understanding of the motions of Mercury, and 
of the relations of its orbit to the plane of the ecliptic, will ren- 
der it easy to understand the same particulars in regard to each 
of the other planets. Standing on the sun, we should see each 
of the planets pursuing a similar course to that of Mercury, all 
moving from west to east, differing from each other chiefly in 
two respects, namely, in their velocities, and in the distances to 
which they recede from the ecliptic, or their inclinations. We 
have supposed the observer to select the plane of the earth's orbit 
as his standard of reference, and to see how each of the other 
orbits is- related to it ; but such a selection of the ecliptic is en- 
tirely arbitrary : the spectator on the sun, who views the motions 
of the planets as they actually exist in nature, would make no 
distinction between the different orbits, but merely inquire how 
they are mutually related to each other. Taking, however, the 
ecliptic as the plane to which all the others are referred, we do 
not, as in the case of the other planets, inquire how its plane is 
inclined, nor what are its nodes, since it has neither inclination 
nor node. 



MOTIONS OF THE PLANETARY SYSTEM. 



219 



357. Such, in general, are the real motions of the planets, and 
such the appearances which the planetary system would exhibit 
to a spectator at the center of motion. But, in order to repre- 
sent correctly the positions of the planetary orbits, at any given 
time, three things must be regarded the Inclination of the orbit 
to the ecliptic the position of the line of the Nodes and the 
position of the line of the Apsides. In our common diagrams, the 
orbits are incorrectly represented, being all in the same plane, as 



Fig. 68. 




in the following diagram, where AEB (Fig. 68) represents the 
orbit of Mercury as lying in the same plane with the ecliptic. 
To exhibit its position justly, AB being taken as the line of the 
nodes, the plane should be elevated on one side about 7, and 
depressed th*e same number of degrees on the other side, turn- 
ing on the line AB as on a hinge. But even then the represen- 
tation may be incorrect in other respects, for we have taken it 
for granted that the line of the nodes coincides with the line 
of the apsides, or that the orbit of Mercury cuts the ecliptic in 
the line AB, the major axis of the orbit, whereas it may lie in 
any given position with respect to the line of apsides, according 



220 THE PLANETS. 

to the longitude of the nodes. If, for example, the line of nodes 
had chanced to pass through Taurus and Scorpio instead of Can- 
cer and Capricorn, then it would have been represented by the 
line ^l instead of the line passing though 55, and the plane when 
elevated or depressed with respect to the plane of the ecliptic, 
would be turned on this line in our figure. Moreover, our dia- 
gram represents the line of apsides as passing through Cancer and 
Capricorn, whereas it may have any other position among the 
signs, according to the longitudes of the perigee and apogee. 

358. Having acquired as correct an idea as we are able of the 
planetary system, as seen from the sun, and of the positions of the 
orbits with respect to the ecliptic, let us next inquire into the na- 
ture and causes of the apparent motions. The apparent motions 
of the planets are exceedingly unlike the real motions, a fact 
which is owing to two causes : first, we view them out of the 
center of their orbits ; secondly, we are ourselves in motion. 
From the first cause, the apparent places of the planets are 
greatly changed by perspective ; and, from the second cause, we 
attribute to the planets changes of place which arise from our 
own motions, of which we are unconscious. 

359. The situation of a heavenly body, as seen from the center 
of the sun, is called its heliocentric place ; as seen from the center 
of the earth, its geocentric place. The geocentric motions of the 
planets must, according to what has just been said, be far more 
irregular and complicated than the heliocentric, as will be evi- 
dent from the following diagram, which represents the geocen- 
tric motions of Mercury for two entire revolutions, embracing a 
period of nearly six months. Let S (Fig. 69) represent the sun, 
1, 2, 3, &c., the orbit of Mercury, a,b,c, &c., that of the earth, and 
GT the concave sphere of the heavens. The orbil^ of Mercury 
is divided into 12 equal parts, each of which he describes in 7j 
days ; and a portion of the earth's orbit described by that body 
in the time that Mercury describes the two complete revolutions, 
is divided into 24 equal parts. Let us now suppose that Mercury 
is at the point 1 in his orbit, when the earth is at the point a ; 
Mercury will then appear in the heavens at A. In 7J days 



MOTIONS OF THE PLANETARY SYSTEM. 



221 



Mercury will have reached 2, while the earth has reached b, 
when Mercury will appear at B. By laying a ruler on the point 
c and 3, d and 4, and so on, in the order of the alphabet, the suc- 
cessive apparent places of Mercury in the 'heavens will be ob- 
tained. From A to C, the apparent motion is direct, or in the 

Fig. 69. 




order of the signs ; from C to G it is retrograde ; at G it is sta- 
tionary a while, and then direct through the whole arc GT. At 
I the planet is again stationary, and afterwards retrograde along 
the arc TX. Hence it appears that the motions of an inferior 
planet, as viewed from the earth, are exceedingly irregular and 
complicated, although it is all the while pursuing its course at a 
nearly uniform rate, and in the same unvarying direction around 
the sun. It moves forward when near the superior conjunction, 



222 THE PLANETS. 

backward when near the inferior, and is stationary near the points 
of greatest elongation. The planet moves sometimes very 
slowly, and then rapidly ; at one time backward over a small space, 
and then forward for a great distance. Yet all these apparent 
irregularities are owing to the two causes already adverted to, 
viz., the effects produced by perspective, and by the motions of 
the spectator himself. Venus exhibits a variety of motions sim- 
ilar to those of Mercury, except that the changes do not succeed 
each other so rapidly, since her period of revolution approaches 

more nearly to that of the earth, 
i 

360. The apparent motions of the superior planets are, like those 
of Mercury and Venus, alternately direct and retrograde, and be- 
tween the two the planets are stationary. In this case, however, 
the earth moves faster than the planet, and the planet has its 
opposition, but no inferior conjunction ; whereas an inferior 
planet has its inferior conjunction, but no opposition. These 
differences render the apparent motions of the superior planets 
in some respect unlike those of Mercury and Venus. On the 
side of the sun most remote from the earth, the motion of a 
superior planet is direct, because, as is the case with Venus in 
her superior conjunction, (see Figure 61,) the only effect of the 
earth's motion is to accelerate it ; but when the planet is in op- 
position, the earth is moving past it with greater velocity, and 
makes the planet seem to move backwards, like the apparent 
backward motion of a vessel when we overtake it and pass rapidly 
by it in a steamboat. 

361. Let ABCD (Fig. 70) represent the earth in different posi- 
tions in its orbit, M a superior planet as Mars, and NR an arc of 
the concave sphere of the heavens. First, suppose the planet to 
remain at rest in M, and let us see what apparent motions it 
would receive from the real motions of the earth. When the 
earth is at B, it will see the planet in the heavens at N ; and as 
the earth moves successively through CDEF, the planet will 
appear to move through OPQR ; B and F are the two points of 
greatest elongation of the earth from the sun, as seen from the 
planet ; between these two points, while passing through the part 



MOTIONS OF THE PLANETARY SYSTEM. 



223 



of its orbit most remote from the planet, (at which time the planet 
is seen in superior conjunction,) the earth, by its own motion, 
gives an apparent motion to the planet in the order of the signs ; 
that is, the apparent motion given by the earth's motion, when 
the planet is seen towards its superior conjunction, is direct. 
But in passing from F to B through A, when the planet is seen 
towards its opposition, the apparent motion given to the planet 
by the earth's motion is retrograde. But the superior planets 




are not in fact at rest, as we have supposed, but are all the while 
moving eastward, though with a slower motion than the earth. 
Indeed, with respect to the remotest planets, as Saturn and Ura- 
nus, the forward motion is so exceedingly slow, that each remains 
for a long time in the same sign of the zodiac. Still, the effect 
of the real motions of all the superior planets eastward, is . to in- 
crease t'he direct apparent motion communicated by the earth, 
and to diminish the retrograde motion, as will be readily seen 
from the figure. 



CHAPTER XI. 

DETERMINATION OF THE PLANETARY ORBITS KEPLER*S DISCOVERIES 

ELEMENTS OF THE ORBIT OF A PLANET QUANTITY OF MAT- 
TER IN THE SUN AND PLANETS. 

362. IN Chapter II. we have shown that the figure of the earth's 
orbit is an ellipse, having the sun in one of the foci, and that the 
earth's radius describes equal spaces in equal times ; and in Chap- 
ter III. we have remarked that these are only particular exam- 
ples under the law of Universal Gravitation, as is also the addi- 
tional fact, that the squares of the periodic times of the planets 
are as the cubes of the major axes of their orbifs. We may now 
learn more particularly the process by which the illustrious Kep- 
ler was conducted to the discovery of these grand laws of the 
planetary system. From the apparent motions of the heavenly 
bodies as seen projected on the face of the sky, the ancient as- 
tronomers inferred that their orbits were necessarily circular, 
arid the motions actually uniform. Still, Hipparchus and Ptolemy 
were not ignorant of the fact, that the sun moves faster through 
the winter than through the summer signs, performing the half 
of his revolution around the earth nearly eight days sooner 
from the autumnal to the vernal, than from the vernal to the 
autumnal equinox. This led them to infer, that the earth is not 
in the center of the circle, but nearer to one side of the circle 
than to the other, by which means the sun would appear to move 
more rapidly in that part of its orbit than in the opposite part, 
just as a steamboat appears to a spectator on the shore to move fast- 
er when nearer than when more remote from the shore, although 
her actual speed is the same in both cases. On a similar suppo- 
sition, Tycho Brahe made a great number of very accurate ob- 
servations on the planetary motions, which served Kepler as 
standards of comparison for results whioh he deduced from cal- 
culations, founded on the application of geometrical reasoning to 



DETERMINATION OF THE PLANETARY ORBITS. 225 

various hypotheses which he successively assumed as to the fig- 
ure of the planetary orbits ; first supposing the orbit to be of a 
certain figure, then determining from the geometrical properties 
of the curve what motions the body would appear to us to have 
when moving in such a path, and finally testing his conclusions 
by comparing them with the facts, as determined by Tycho, from 
observation. 

\ 

363. Kepler first applied himself to investigate the figure of the 
orbit of Mars, the motions of which planet appeared more irreg- 
ular than those of any other planet except Mercury, which, being 
seldom seen, had been very little studied. Like Ptolemy and 
Tycho, he first supposed the orbit to be circular, and the planet 
to move uniformly about a point at a certain distance from the 
sun. He made seventy suppositions befoi'e he obtained one that 
agreed with observation, the calculation of which was extremely 
long and tedious, occupying him more than five years.* The 
supposition of an equable motion in a circle, however varied, 
could not be % made to conform to the observations of Tycho, 
whereas the supposition that the orbit was an oval figure, de- 
pressed at the sides, but coinciding with a circle at the perihelion, 
agreed so nearly with observation as to leave no doubt that the 
orbit of Mars is an ellipse, having the sun in one of its foci. He 
immediately inferred that the same is true of the orbits of all the 
other planets ; and a similar comparison of this hyppthesis with 
observation, confirmed its truth. Thus he established the first 
great law, viz., The planets revolve about the sun in ellipses, hav- 
ing the sun in one of the foci. 

364. Kepler also discovered from observation, that the veloci- 
ties of the planets, when in their apsides, are inversely as the 
distances respectively, and therefore the product of the velocity 



* Logarithms were invented during the age of Kepler, but were not available 
to him until his most laborious calculations had been performed. In relation to 
these, he expresses himself thus : Si te hujus laboriosce mcthodi pertccsum fueritjure 
tnei te misereat, qui earn ad minimum septuagies ivi cum plurima temporis jactura ; 
st mirari desines hunc quintumjam annum abire, ex quo Martem aggressus sum. 

29 



226 THE PLANETS. 

into the distance is a constant quantity, as was proved of the 
solar orbit, (Art. 167.) From this it follows that the radius vec- 
tor in each case describes equal areas in equal times, since the 
product of a triangle or circular sector into the perpendicular, is 
a measure of its area. But in this case the base is the space 
described by the planet in a given time, and the perpendicular is 
the radius vector, (Fig. 32, p. 86.) Although he could not prove, 
from observation, that the same was true in every point of the 
orbit, yet analogy suggested that such was probably the fact. 
Therefore, assuming this principle as true, and hence deducing 
the equation of the center, (Art. 200,) he found the result to 
agree with observation, and therefore concluded in general, that 
the radius vectors of the planetary orbits describe about the sun 
equal areas in equal times. 

365. Having in his researches, that led to the discovery of 
the first of the above laws, found the relative mean distances 
of the planets from the sun, (Art. 308,) and, knowing their 
periodic times from observation, Kepler next endeavored to 
ascertain if there was any relation between the distances and 
times of revolution, having a strong passion for tracing analogies 
in nature. He saw at once that the more distant a planet is from 
the sun, the slower it moves ; so that the periodic times of the 
remoter planets are increased on two accounts ; first, because 
they have aJonger path to traverse ; and secondly, because they 
actually move more slowly in their orbits than the planets nearer 
the sun. Saturn, for example, is 9|- times further from the sun 
than the earth is ; and since the circumferences of circles are as 
their radii, the orbit of Saturn must be larger than the earth's in 
the same ratio ; so that if the periodic time of Saturn were longer 
than the earth's merely because its orbit is larger, that period 
would be 9^ years, whereas it is 30 years. Hence it is evident, 
that the periodic times of the planets increase in a greater ratio 
than their distances from the sun, but in a less ratio than the 
squares of the distances, for then the time of Saturn would be 
about 90 years. Kepler then compared the squares of the times 
with the cubes of the distances, and found an exact agreement 
between them. Thus he discovered the famous law, the squares 



ELEMENTS OF THE PLANETARY ORBITS. 227 

of the periodic times of all the planets, are as the cubes of their 
mean distances from the sun* 

366. This law is strictly true only in relation to planets whose 
quantity of matter in comparison with that of the central body is 
inappreciable. When this is not the case, the periodic time is 
shortened in the ratio of the square root of the sun's mass divi- 
ded by the sun's plus the planet's mass, as expressed by the 

formula [^ 1 -. The mass of the planets is, however, so 

\M + mf 

small compared to the sun's, that this modification of the law is 
unnecessary except where extreme accuracy is required. 

ELEMENTS OF THE PLANETARY ORBITS. 

367. The particulars necessary to be known in order to deter- 
mine the precise situation of a planet at any instant, are called 
the Elements of its Orbit. They are seven in number, of which 
the first two determine the absolute situation of the orbit, and the 
other five relate to the motion of the planet in its orbit. These 
elements are, 

(1.) The position of the line of the nodes. 

(2.) The inclination to the ecliptic. 

(3.) The periodic time. 

(4.) The mean distance from the sun, or semi-axis major. 

(5.) The eccentricity. 

(6.) The place of the perihelion. 

(7.) The place of the planet in its orbit at a particular epoch. 

368. It may at first view be supposed that we can proceed to 
find the elements of the orbit of a planet in the same manner as 
we did those of the solar or lunar orbit, namely, by observations 
on the right ascension and declination of the body, converted into 
latitudes and longitudes by means of spherical trigonometry, (See 
Art. 132.) But in the case of the moon, we are situated in the 
center of her motions, and the apparent coincide with the real 
motions ; and, in respect to the sun, our observations on his appa- 
rent motions give us the earth's real motions, allowing 180 differ- 

* Vince's Complete System, L 98. 



228 THE PLANETS. 

ence in longitude. But as we have already seen, the motions of 
the planets appear exceedingly different to us, from what they 
would if seen from the center of their motions. It is necessary 
therefore to deduce from observations made on the earth the cor- 
responding results as they would be if viewed from the center of 
the sun ; that is, in the language of astronomers, having the geo- 
centric place of a planet, it is required to find its heliocentric place. 

369. The first steps in this process are the same as in the case 
of the sun and moon. That is, for the purpose of finding the right 
ascension and declination, the planet is observed on the meridian 
with the Transit Instrument and Mural circle, (See Arts. 155 and 
230,) and from these observations, the planet's geocentric longi- 
tude and latitude are computed by spherical trigonometry. The 
distance of the planet from the sun is known nearly by Kepler's 
law. From these data it is required to find the heliocentric lon- 
gitude and latitude. 

Let S and E (Fig. 71) be the sun and earth, P the planet, PO 
a line drawn from P perpendicular to the ecliptic, SA the direc- 



Fig. 71. 




tion of Aries, and EH parallel to SA, ahd therefore (on account 
of the immense distance of the fixed stars) also in the direction 
of Aries. Then OEH, being the apparent distance of the planet 
from Aries in the direction of the ecliptic, is the geocentric longi- 
tude, and OEP, being the apparent distance of the planet from the 
ecliptic taken on a secondary to the ecliptic, is the geocentric 
latitude. It is obvious also that the angles OSA and PSO are 



ELEMENTS OF THE PLANETARY ORBITS. 229 

the heliocentric longitude and latitude. The planet's angular dis- 
tance from the sun, PES, is also known from observation. Hence, 
in the triangle SEP, we know SP and SE and the angle SEP, from 
which we can find PE ; and knowing PE and the angle PEO, we 
can find OE, since OEP is a right angled triangle. Hence in the 
triangle SEO, ES and EO, and the angle SEO (=OEH-SEH= 
difference of longitude of the planet and the sun) are known, and 
hence we can obtain OSE, which added to the sun's longitude 
ESA,* gives us OSA the planet's heliocentric longitude. 
Also, because PS : Rad. : : OP : Sin. PSO. 

.-. PS x Sin. PSO=OPxRad. 
But EP : Rad. : : OP : Sin. OEP. 

.-. EPxSin. OEP=OPxRad. 

.-. PS x Sin. PSO=EPxSin. OEP. 

.-. PS : EP : : Sin. OEP : Sin. PSO. 

The first three terms of this proposition being known, the last 
is found, which is the heliocentric latitude.^ 

370. Having now learned how observations made at the earth 
may be concerted into corresponding observations made at the 
sun, we may proceed to explain the mode of finding the several 
elements before enumerated ; although our limits will not permit 
us to enter further into detail on this subject, than to explain the 
leading principles on which each of these elements is determined. J 

371. First, to determine the position of the Nodes, and the In- 
clination of the Orbit. 

These two elements, which de- Fig- 72 - 

termine the situation of the orbit, 
(Art. 367,) may be derived from two 
heliocentric longitudes and latitudes. ^ 

Let AR and AS (Fig. ^2) be two 

i 

* Strictly, ESA, being the supplement of the angle SEH, is the supplement of 
the eun's longitude. 

f Brinkley's Elements of Astronomy, p. 164. 

\ Most of these elements admit of being determined in several different ways, an 
explanation of which may be found in the larger works on Astronomy, as Vince's 
Complete System, Vol. I. Gregory's Ast. p. 212. Woodhouse, p. 562. 




230 THE PLANETS. 

heliocentric longitudes, PR and QS the heliocentric latitudes, 
and N the ascending node. Then, by Napier's theorem, 
(Art. 132,) 

Sin. NR (=AR-AN)_ plvrT? _sm. NS (=AS-AN) 
"talTPR tan. QS 

.-.Sin. ARxcos. AN cos. ARx-sin. AN*_ 

tan. PR 
sin. AS x cos. AN cos. AS x sin. AN 



But tan. AN^ 



tan. QS 
sin. AN Sin. ARxtan. QS-sin. AS x tan. PR 



cos. AN Cos. AR x tan. QS cos. AS x tan. PR' 
But AN is the longitude of the ascending node ; and its value 
is found in terms of the heliocentric longitudes and latitudes pre- 
viously determined, (Art. 369.) 

Again, since AN is found, we may deduce from the first 
equation above the value of PNR, which is the inclination of the 
orUt.-\ 

372. Secondly, to find the Periodic Time. 

This element is learned, by marking the interval that passes 
from the time when a planet is in one of the nodes ijptil it returns 
to the same node. We may know when a planet is at the node, 
because then its latitude is nothing. If, from a series of observa- 
tions on the right ascension and declination of a planet, we deduce 
the latitudes, and find that one of the observations gives the lati- 
tude 0, we infer that the planet was at that moment at the node. 
But if, as commonly happens, no observation gives exactly 0, then 
we take two latitudes that are nearest to 0, but on opposite sides 
of the ecliptic, one south and the other north, and as the sum of the 
arcs of latitude is to the whole interval, so is one of the arcs to the 
corresponding time in which it was described, which time being 
added to the first observation, or subtracted from the second, will 
give the precise moment when the planet was at the node. 

By repeated observations it is found, that the nodes of the plan- 
,ets have a very slow retrograde motion. 

373. If the orbit of a planet cut the ecliptic at right angles, then 
* Day's Trig. Art. 208. f Brinkley, p. 166. 



ELEMENTS OF THE PLANETARY ORBITS. 231 

small changes of place would be attended by appreciable differ- 
ences of latitude ; but in fact the planetary orbits are in general 
but little inclined to the ecliptic, and some of them lie almost in 
the same plane with it. Hence arises a difficulty in ascertaining 
the exact time when a planet reaches its node. Among the most 
valuable observations for determining the elements of a planet's 
orbit, are those made when a superior planet is in or near its op- 
position to the sun, for then the heliocentric and geocentric lon- 
gitudes are the same. When a number of oppositions are 
observed, the planet's motion in longitude, as would be observed 
from the sun, will be known. The inferior planets also, when in 
superior conjunction, have their geocentric and heliocentric lon- 
gitudes the same. When in inferior conjunction, these lon- 
gitudes differ 180 ; but the inferior planets can seldom be 
observed in superior conjunction, on account of their proximity 
to the sun, nor in inferior conjunction except in their transits, 
which occur too rarely to admit of observations sufficiently nu- 
merous. Therefore, we cannot so readily ascertain by simple 
observation, the motions of the inferior planets seen from the sun, 
as we can those of the superior.* 

374. Hence, in order to obtain accurately the periodic time of 
a planet, we find the interval elapsed between two oppositions 
separated by a long interval, 'when the planet was nearly in the 
same part of the zodiac. From the periodic time, as determined 
approximately by other methods, it may be found when the planet 
has the same heliocentric longitude as at the first observation. 
Thus the time of a complete number of revolutions will be 
known, and thence the time of one revolution. The greater the 
interval of time between the two oppositions, the more accurately 
the periodic time will be obtained, because the errors of observa- 
tion will be divided between a great number of periods ; there- 
fore by using very accurate observations, much precision may be 
attained. For example, the planet Saturn was observed in the 
years 228 B. C., March 2, (according to our reckoning of time,) 
to be near a certain star called 7 Virginis, and it was at the same 

* Briukley, p. 167. 



232 THE PLANETS. 

time nearly in opposition to the sun. The same planet was again 
observed in opposition to the sun, and having nearly the same 
longitude, in Feb. 1714. The exact difference between these 
dates was 1943y. 1 18d. 21h. 15m. It is known from other sources, 
that the time of a revolution is 29J years nearly, and hence it was 
found that in the a'bove period there were 66 revolutions of Sat- 
urn; and dividing the interval by this number, we obtain 29*444 
years, which is nearly the periodic time of Saturn according to 
the most accurate determination. 

375. Thirdly, to determine the distance from the sun, and majo? 
axes of the planetary orbits. 

The distance of the earth from the sun being known, the mean 
distance of any planet (its periodic time being known) may be 
found by Kepler's law, that the squares of the periodic times are 
as the cubes of the distances. The method of finding the dis- 
tance of an inferior planet from the sun by observations at the 
greatest elongation, has been already explained, (see Art. 308.) 
The distance of a superior planet may be found from observations 
on its retrograde motion at the time of opposition. The periodic 
ttmes of two planets being known, we of course know their mean 
angular velocities, which are inversely as the times. Therefore, 
let Ee (Fig. 73) be a very small portion of the earth's orbit, and 
M??z a corresponding portion of that of a superior planet, de- 
scribed on the day of opposition, about the sun S, on which day 

Fig. 73. 



M 



the three bodies lie in one straight line SEMX. Then the angle 
ESe and MSm, representing the respective angular velocities of 
the two bodies are known. Now if em be joined, and prolonged 
to meet SM continued in X, the angle EXe, which is equal to the 
alternate angle Xey, being equal to the retrogradation of the planet 
in the same time, (being known from observation,) is also given. 
Ee, therefore, and the angle EXe being given in the right-angled 
triangle EXe, the side EX is easily calculated, and thus SX be- 
comes known. Consequently, in the triangle SwzX, we have 



ELEMENTS OF THE PLANETARY ORBITS. 233 

given the side SX, and the two angles wSX and wXS, whence 
the other sides Sm and mX are easily determined. Now Sm is 
the radius of the orbit of the superior planet required, which, in 
this calculation, is supposed circular, as well as that of the earth, 
a supposition not exact, but sufficiently so to afford a satisfac- 
tory approximation to the dimensions of its orbit, and which, if 
the process be often repeated, in every variety of situation at 
which the opposition can occur, will ultimately afford an average 
or mean value of its distance fully to be depended on.* 

376. Fourthly, to determine the place of the perihelion the 
epoch of passing the perihelion and the eccentricity. 

An easy method of finding the place of the perihelion, and 
of course the position of the line of the apsides, of a planetary 
orbit, and the eccentricity, is the following. From a series of 
observations on the greatest elongations of a planet from the 
sun, we shall find one that is a minimum, and another that is a 
maximum. The former denotes the place of the perihelion, the 
latter of the aphelion. Thus, (Fig. 60,) if in a long series of ob- 
servations on the greatest elongations of Mercury, the value of 
SB were at any time to be the least of all, we should know that 
that point is the place of the perihelion, and of course the point 
diametrically opposite is the place of the aphelion. Moreover, by 
calculating the relative distances of the planet from the sun at 
these two points, as in Art. 308, we ascertain the length of the 
least and the greatest radius vector, and half the difference of 
these two lines constitutes the eccentricity. This method, how- 
ever, is applicable only to the inferior planets Mercury and Ve- 
nus. The place of the nodes also can be determined by a series 
of observations on the latitudes of a planet, being at those points 
where the latitude is nothing. In most cases, indeed, the geo- 
centric would be different from the true heliocentric latitude, 
and of course observation would not give the exact positions of 
the nodes ; but when, as is sometimes the case, the planet is in 
conjunction or in opposition at the time of passing the node, then 
it is seen in the same place as if viewed from the sun : the geo- 

* Sir J. HerscheL 
30 



234 



THE PLANETS. 



Fig. 74. 



centric coincides with the heliocentric place, and the observed 
is the true place of the node. 

. But trigonometry, building on a few instrumental observations, 
affords other modes of arriving at the elements of a planetary 
orbit, one of which is derived from the greatest equation of the 
center, (Art. 200.) For since the two points in the orbit where 
this becomes greatest- are equally distant from the apsides, by 
bisecting the interval between these 
two points, we obtain the position of 
the perihelion and aphelion. Let 
AEBF (Fig. 74) be the orbit of the 
planet, having the sun in the focus at 
S. In an ellipse, the square root of 
the product of the semi-axes gives 
the radius of a circle of the same 
area as the ellipse.* Therefore, 
with the center S, at the distance 
SE= v/AKx OK, describe the circle 
CEGF, then will the area of this cir- 
cle be equal to that of the ellipse. At 
the same time that a body departs 
from- A the aphelion, let a body begin to move with a uniform 
motion from C through the periphery CEGF, and perform a whole 
revolution in the same period that the planet describes the ellipse ; 
the motion of this body will represent the equable or mean motion 
of the planet, and it will describe around S areas or sectors of 
circles which are proportional to the times, and equal to the ellip- 
tic areas described in the same time by the planet. Let the 
equable motion, or the angle about S proportional to the time, be 
CSM, and take ASP equal to the sector CSM ; then the place of 
the planet will be P ; MSC will be the mean anomaly, (Art. 200,) 
DSC the true anomaly, and MSB the equation of the center. 
Since the sectors CSM and ASP are equal, and the part CSD is 
common to both, PACD and MSD are equal ; and therefore 
PACD is the measure of the equation of the center, which is 
greatest when PACD becomes ACE, that is, at the point where 




* Day's Mensuration. 



ELEMENTS OF THE PLANETARY ORBITS. , 235 

- 

the ellipse and the circle intersect one another. For since the 
semi-ellipse AEB and the semicircle CEG are equal, the planet, 
starting from the aphelion A, will at first fall behind the body 
moving in the circle, and will not overtake it till it arrives at B. 
Taking from the ellipse and the circle the common part CEB, 
the remainders AEC and BEG are equal. The true anomaly 
equals ASm, the mean anomaly ASV, and the difference or 
mSV equals the equation of the center. But since ACE=GBE, 
therefore GBE + CSR = CSR + ERm+mSV /. GBE-ERi= 
mSV. Hence the equation of the center becomes less than GBE 
or ACE after passing the point E, and consequently, the equation 
of the center is greatest at the point E, where the real motion of 
the planet is equal to its mean motion. The mean motion for 
any given time is easily found ; for the time of revolution is to 
360 as the given time is to the number of degrees for that time. 
Observation shows when the actual motion of the planet is the 
same with this. Now, the equation of the center is greatest 
twice in the revolution, on opposite sides of the orbit, as at E and 
F, which points lie at equal distances from the apsides A and B ; 
and since the whole arc EAF or EBF is known from the time 
occupied in describing it, therefore, by bisecting the arc, we find 
the points A and B, the aphelion and perihelion, and, consequently, 
the position of the line of the apsides. The time of describing 
the area EBF -being known, by bisecting this interval we obtain 
the moment of passing the perihelion, which gives us. the place 
of the planet in its orbit at a particular epoch* 

377. The amount of the greatest equation evidently depends 
on the eccentricity of the orbit, since it arises wholly from the 
departure of the ellipse from the figure of a perfect circle ; hence 
the greatest equation affords the means of determining the eccen- 
tricity itself. In orbits of small eccentricity, as is the case with 
most of the plar dtary orbits, it is found that the arc which meas- 
ures the greatest equation is very nearly equal to the distance 
between the foci, which always equals twice the eccentricity, the 
measure of the eccentricity being the distance from the focus to 

* Gregory's Astronomy, p. 197. 



236 THE PLANETS. 

the center of the ellipse. The angular value of radius is 57 17' 
44". 8; for, 

3.14159 : 1 : : 180 : 57 17' 44".8. 

Therefore, 57 17' 44". 8 : radius : : half the greatest equation 
of the center : the eccentricity * 

The foregoing explanations of the methods of finding the ele- 
ments of the orbits, will serve jn general to show the learner how 
these particulars are or may be ascertained : yet the methods 
actually employed are usually more refined and intricate than 
these. In astronomy, scarcely an element is presented simple 
and unmixed with others. Its value when first disengaged, must 
partake of the uncertainty to which the other elements are sub- 
ject, and can be supposed to be settled to a tolerable degree of 
correctness, only after multiplied observations and many revi- 
sions, f Indeed, a large part of the most arduous labors of astron- 
omers have been employed in finding the elements of the plane- 
tary orbits, with the wonderful degree of precision which has 
finally been attained. 



aUANTITY OF MATTER IN THE SUN AND PLANETS. 

378. It would seem at first view very improbable, that an in- 
habitant of this earth would be able to weigh the sun and planets, 
and estimate the exact quantity of matter which they severally 
contain. But the principles of Universal Gravitation conduct 
us to this result, by a process remarkable for its simplicity. By 
comparing the relations of a few elements that are known to us, 
we ascend to the knowledge of such as appeared beyond the pale 
of human investigation. We learn the quantity of matter in a 
body by the force of gravity it exerts. Let us see how this force 
is ascertained. 

379. The quantities of matter in two bodies, may be found in 
terms of the distances and periodic times of two bodies revolving 
around them respectively, being as the cubes of the distances divi- 
ded by the squares of the periodic times. 

* Vince's Complete System, 1. 118. f "Woodhouse, p. 579. 



QUANTITY OF MATTER IN THE SUN AND PLANETS. 237 



The force of gravity G in a body whose quantity of matter is 
M and distance D, varies directly as the quantity of matter, and 

inversely as the square of the distance ; that is, G <r =-. But it 

is shown by writers on Central Forces, that the force of gravity 
also varies as the distance divided by the square of the periodic 

D* M D D 3 

time, or G oc . Therefore, ^o: . , and M oc p. Thus we 

may find the respective quantities of matter in the earth and the 
sun, by comparing the distance and periodic time of the moon, re- 
volving around the earth, with the. distance and periodic time of 
the earth revolving around the sun. For the cube of the moon's 
distance from the earth divided by the square of her periodic time, 
is to the cube of the earth's distance from the sun divided by the 
square of her periodic time, as the quantity of matter in the earth 

. 238,54B 3 95,000,000 s 
is to that in the sun. That is, ^r : " : : * :353 ' 385 ' 



The most exact determination of this ratio, gives for the mass 
of the sun 354,936 times that of the earth. Hence it appears 
that the sun contains more than three hundred and fifty-four 
thousand times as much matter as the earth. Indeed, the sun 
contains eight hundred times as much matter as all the planets. 

Another method, well suited to popular illustration, of weigh- 
ing the earth against the sun, is the following. Knowing the 
radii of the solar and lunar orbits respectively, we can easily find 
the space which the moon descends towards the earth, and the 
earth towards the sun, in any given time, as an hour. Thus, 
(Fig. 75,) if we know the radius AE of the orbit, we can deter- 
mine the length of the arc Ab, described in an hour, and also the 
length of the hypothenuse BE. But BE AE=B6, the space 
through which the central attracts the revolving body in the 
given time." The earth draws the moon towards itself about 11 
miles per hour, and the sun^ draws the earth towards itself 24.4 
miles per hour ; that is, the sun exerts a force 2} greater on the 
earth than the earth does on the moon. But were the sun at the 
same distance as the moon, his force of attraction would be the 

* Olmsted's Natural Philosophy, Art. 185. 



238 






THE PLANETS. 



square of 400, or 160,000 times as great as it is now ; that is, it 
would be 2Jx 160,000 times as great as the earth's attraction, 
and, consequently, must have 2jx 160,000=352,000 times as 




much matter, a result agreeing nearly with the former. The 
agreement would be exact if more precise numbers were em- 
ployed, but our object is here merely to illustrate the method. 

380. The mass of each of the other planets that have satellites 
may be found, by comparing the periodic time of one of its sat- 
ellites with its own periodic time around the sun. By this means 
we learn the ratio of its quantity of matter to that of the sun. 
The masses of those planets which have no satellites, as Venus 
or Mars, have been determined, by estimating the force of at- 
traction whteh they exert in disturbing the motions of other 
bodies. Thus, the effect of the moon in raising the tides, leads 
to a knowledge of the quantity of matter in the moon ; and the 
effect of Venus in disturbing the motions of the earth, indicates 
her quantity of matter.* 

381. The quantity of matter in bodies varies as their magni- 
tudes and densities conjointly. Hence, their densities vary as 



* These estimates are made by the most profound investigations in Laplace's 
Me'canique Celeste, Vol. III. 



QUANTITY OF MATTER IN THE SUN AND PLANETS. 239 

their masses divided by their magnitudes ; and since we know 
the magnitudes of the planets, and can compute as above their 
masses, we can thus learn their densities, which, when reduced 
to a common standard, give us their specific gravities, or show 
us how much heavier they are than water. Worlds, therefore, 
are weighed with almost as much ease as a pebble, or an article 
of merchandise. 

The densities and specific gravities of the sun, moon, and 
planets, are estimated as follows :* 





Density. 


Specific Gravity. 


Sun, 


0.25 


1.37t 


Moon, . 


0.56 


1 

3.27 


Mercury, 


1.12 


6.13 


Venus, . 


0.92 


5.04 


Earth, . 


1.00 


5.48 


Mars, 


0.95 


5.20 


Jupiter, . 


0.24 


1.31 


Saturn, . 


0.14 


0.76 


Uranus, . 


0.24 


1.31 


Neptune, 


0.14 


0.76 



From this table it appears that the sun consists of matter but 
little heavier than water ; but that the moon is more than three 
times as heavy as water, though less dense than the earth, which 
is five and a half times heavier than water. It also appears 
that the planets near the sun are, as a general fact, more dense 
than those more remote, Mercury being as heavy as many of 
the metallic ores, while Saturn is as light as a cork. The 
decrease of density, however, is not entirely regular, since Venus 
is a little lighter than the earth, and Saturn than Uranus. 

* Hersohel. 

f The earth being taken, according to Baily, at 5.48, the specific gravities of the 
other bodies (which are found by multiplying the density of each by the specific 
gravity of the earth) are here stated somewhat higher than they are given in 
most works 



CHAPTER XII. 

t* 

PERTURBATIONS OF THE PLANETS STABILITY OF THE SYSTEM 

NUMERICAL RELATIONS OF THE PLANETS PROBLEMS. 

382. THE perturbations occasioned in the motions of the plan- 
ets by their action on each other are very numerous, since every 
body in the system exerts an attraction on every other, in confor- 
mity with the law of universal gravitation. Venus and Mars, 
approaching as they do at times comparatively near to the earth, 
sensibly disturb its motions ; and Jupiter and Saturn, although 
very far asunder, still, in consequence of their great masses, 
exert on each other, when on the same side of the heavens es- 
pecially, a decided influence. Moreover, the sun, by his unequal 
action on the several planets, in consequence of the peculiar fig- 
ure of each, produces various irregularities in their motions. As 
in the case of the earth and moon, (Art. 243,) these perturbations 
are divided into periodical and secular : periodical, when com- 
pleted in comparatively short periods, as those, for example, 
which undergo all their changes during one revolution of the 
planet ; and secular, when completed only in very long periods, 
as those which affect the form and inclination of the orbits. 

383. If the only bodies in the system were a central body like 
the sun, and a revolving body like Venus, then, when the planet 
was once put in motion with such a projectile force as to make 
it describe an ellipse, it would forever continue to describe the 
same figure without the least variation, the radius vector always 
passing over equal spaces in equal times ; but now introduce a 
third body so near as to exert on it a decided attraction, and its 
motions no longer retain their simplicity, but become complica- 
ted by the conflicting influences of the two attracting bodies. 
The sun, however, in consequence of its mass, which is eight 
nundred times as great as that of all the planets, and, of course, 



PERTURBATIONS OF THE PLANETS. 241 

vastly greater than that of any one of them, exerts a force so 
much superior to that of any or all the other disturbing bodies, 
that the elliptical figure of the orbits is nearly maintained, and a 
near approximation to the place of a pdanet is obtained, by neg- 
lecting all those minor forces, and si aply contemplating it as re- 
volving in an elliptical orbit. Still it is essential, in order to find 
the exact place of a planet at any given time, that all these irreg- 
ularities, minute as they may be, be carefully summed up, and 
their resultant applied to the elliptical motions. To investigate 
these perturbations, to estimate their precise amount, and to reg- 
ister them in tables, for the use of the practical astronomer, have 
constituted a large part of the labors of modern astronomy. The 
knowledge gained by astronomers of the planetary motions, con- 
sidering the very numerous irregularities, both periodical and 
secular, to which they are subject, is truly wonderful. The mo- 
tion of Jupiter, for instance, is so perfectly calculated, that 
astronomers have computed ten years beforehand the time at 
which it will pass the meridian of different places, and we find 
the prediction correct within half a second of time.* The 
more obvious irregularities have been detected by observation ; 
the more minute, by following out the consequences of universal 
gravitation. Even those at first revealed to the instruments of 
the astronomer, have been confirmed and estimated with greater 
accuracy, by the same far-reaching principle ; and many of the 
irregularities have been first brought to. light by this theory, 
which had before eluded observation ; although, when once 
pointed out as a result of the principle of gravitation, careful 
instrumental measurements have confirmed them, except in cases 
where the force was too minute to be reached by the most re- 
fined observation. Periodical perturbations among the bodies of 
the solar system, may be compared to the regular flux and reflux 
of the tides, by which the ocean daily oscillates about its mean 
level, without any permanent change of level, while secular per- 
turbations would resemble any slow changes of level, which, accu- 
mulating from time to time, might finally become obvious to 
measures of the depths of the ocean, as recorded from age to age. 

* Airy. 
31 



242 THE PLANETS. 

As an example of the extreme minuteness of some of these secular 
perturbations, we may instance the changes in the eccentricity 
of the earth's orbit. Thg entire eccentricity is so small that the 
figure, when drawn on paper in just proportions, can scarcely be 
distinguished from a circle, the focus of the ellipse being distant 
from the center only about -^ part of the semi-major axis. But 
the change of eccentricity in a century, is only the twenty-five 
thousandth part of the whole, or the fifteen hundred thousandth 
part of the semi-major axis. 

384. But although the secular inequalities of the planetary mo- 
tions are exceedingly slow, yet may they not in time accumulate 
so as to derange the whole system ; and do they not at least indi- 
cate that the system carries within it the seeds of its own dissolu- 
tion ? So far is this from being the case, that the stability of the 
solar system is a fact established on the most satisfactory evidence, 
and its demonstration is among 'the finest triumphs of physical 
astronomy. Even a superficial view of the system will convince 
us that care has been bestowed on this point by several obvious 
arrangements. One is, that the planets have severally so small 
masses compared with the sun, as to interfere but little, at most, 
with the supremacy of his control over the planetary motions. 
Another is, that the planets are placed at such great distances from 
each other,- a distance which is greater among the largest bod- 
ies, as Jupiter and Saturn, than among the smaller, as the earth 
and Venus ; and another still, that the orbits are less eccentric 
when the masses of the bodies are greater, by which provision 
they are always maintained at a remote distance from the sun. 
Were the orbit of Jupiter as eccentric as that of Mars, he would 
approach so near the earth at his perihelion, as greatly to endanger 
its stability. But if even these general considerations might 
convince us that the stability of the solar system is provided for, 
a more profound investigation will reveal this truth in a far more 
admirable light. This object is especially secured by the follow- 
ing remarkable provisions. 

First, by the invariability of the grand axes, and of the peri- 
odic times ; secondly, by the fact, that whatever irregularities a 
planet undergoes on one side of its orbit, (so far as respects the 



STABILITY OF THE SYSTEM. 243 

periodical perturbations,) they are compensated on the other 
side ; so that, when it returns to a given point, as the node or 
the perihelion, any irregularities it may have felt in different parts 
of its orbit, neutralize one another, and therefore do not consti- 
tute an accumulating mass of errors ; and, thirdly, by this, that 
all the secujar perturbations are restricted within narrow limits, 
oscillating to and fro ; but, before they can proceed so far on one 
side as to endanger the stability of the system, they turn about 
and proceed, for a similar period, in the opposite direction. 

385. These truths have been established by the most rigorous 
mathematical demonstrations, by the successive labors of three 
very celebrated mathematicians, Euler, Lagrange, and Laplace. 
It \vas demonstrated that the major axes of the planetary orbits, 
and the times of their revolutions around the sun, are subject to no 
secular perturbations, nor to any variation whatever, but such as, 
in the course of a single revolution, exactly compensate and neu- 
tralize' each other. This is a most important point in relation to 
the stability of the system ; for if the lengths of the major axes 
varied, then, of course, the times of revolution would vary, 
(since, by Kepler's 3d law, the squares of the periodic times are 
in a constant ratio to the cubes of the major axes,) and we 
should have years of unequal length, and the earth, by approach- 
ing at one time nearer to the sun, and at another receding fur- 
ther from it, would render the changes of temperature too great 
for the existence of animal or vegetable life ; and similar evils, 
it is probable, would result to the economy of the other planets. 
It was next established, that the eccentricities of the planetary 
orbits, although they have been undergoing constant changes in 
all time past, and will continue to undergo them* in all future 
ages, can never vary beyond a certain moderate limit, entirely 
within the bounds of safety to the stability of the system. The 
eccentricity of the earth's orbit, for example, has been diminish- 
ing from the creation of the world ; and although, as we have 
seen, the rate of diminution is exceedingly slow, yet, in the pro- 
gress of centuries, it would totally change the character of the 
earth's orbit ; first reducing it to the circular form, and finally 
carrying its eccentricity to a fatal extreme. In like manner, the 



244 THE PLANETS. 

inclination of the earth's orbit to the equator is constantly di- 
minishing, and is now about two-fifths of a degree less than it 
was in the days of Aristotle ; and, were this to proceed in the 
same direction, the equator and ecliptic would coincide, the 
change of seasons would cease, and the whole economy of na- 
ture would be subverted. But Laplace has demonstrated, that 
such an event can never occur, nor can the entire extent of this 
variation exceed three degrees. It is worthy of remark, that 
those perturbations, such as changes in the place of the peri- 
helion, affecting a change of direction in space of the major axis 
of the orbit, or in the place of the nodes, which, by accumulating, 
do not endanger the stability of the system, proceed onward 
through the entire circuit of the heavens, while perturbations 
which, by indefinite accumulation, would bring ruin to the sys- 
tem, such as variations of eccentricity and of inclination, are not 
progressive, but oscillatory, waving to and fro within the limits 
of entire safety. 

386. These great ends would not have been secured, had the 
system been constructed differently from what it is. Numerous 
conditions must concur in order to produce these results ; the 
mass of the sun must have greatly exceeded that of any or -all the 
planets ; the eccentricities of the orbits must have been small ; 
and the planets must all have revolved around the sun in the 
same direction, and in planes but little inclined to each other.* 
It was also necessary that the periodic times of the planets should, 
in general, be incommensurable ; for were their periods such that 
one planet would revolve a certain number of times exactly, 
while another planet, next to it, revolved a certain other even 
number of times, then, when they once came into the sphere of 
each other's influence, they might remain under it so long, 
and return to their relative position so often, as seriously to de- 
range their orbits. An instance of this, in fact, occurs in the 
case of Jupiter and Saturn, five revolutions of Jupiter being 
nearly equal to two of Saturn, a relation which gives rise to 
what is called the long inequality of Saturn and Jupiter. Similar 

* Laplace, Sys. du Monde. Herschel's Outlines. Grant's History of Physical 
Astronomy. Pontecoulant's Trait. Elemen. de Phys. Celeste. 



RELATIONS BETWEEN BODIES OF THE SOLAR SYSTEM. 245 

effects result from a near commensurability of the mean motions 
of any other two planets. One exists between the earth and Ve- 
nus, 13 times the period of Venus being very nearly equal to 8 
times that of the earth ; still the influence of this disturbing cause 
is so nicely compensated, and its effects so distributed, that, ac- 
cording to Mr. Airy, (who was the first to detect it,) it amounts, 
at its maximum, to no more than a few seconds for a period of 
240 years. The laws which regulate the eccentricities and in- 
clinations of the planetary orbits, (says an able writer on Physical 
Astronomy,) combined with the invariability of the mean distan- 
ces, secure the .permanence of the solar system throughout an 
indefinite lapse of ages, and offer to us an impressive indication 
of the Supreme Intelligence which presides over nature, and per- 
petuates her beneficent arrangements. When contemplated 
merely as speculative truths, they are unquestionably the most 
important which the transcendental analysis has disclosed to the 
researches of the geometer ; and their complete establishment 
would suffice to immortalize the names of Lagrange and La- 
place, even although these great geniuses possessed no other 
claims to the recollection of posterity.* 

NUMERICAL RELATIONS BETWEEN THE BODIES OF THE SOLAR 

SYSTEM. f 

387. If we contemplate the relations subsisting between a cen- 
tral body, as the sun, and a revolving body, as one of the planets, 
it will be readily understood, that if the quantity of matter in the 
central body is increased, while the distance of the revolving 
body remains the same, the velocity of the revolving body must 
te increased also, in order to generate a sufficient centrifugal 
force to counterbalance the increased force of attraction in the 
central body, arising from the increase of its mass ; and that, 
were the force of attraction diminished by removing the body to 
a greater distance from the center, then the rate of its motion 

* Grant's Hist. Phys. Ast.p. 66. 

f In the preparation of this article, the author has derived much assistance from 
a small work, now nearly out of print, containing the substance of three lectures 
delivered to the students of Yale College in 1781, by Rev. Nehemiah Strong, at 
that time Professor of Mathematics and Natural Philosophy. 



246 THE PLANETS. 

would also have to be diminished, otherwise the centrifugal force 
would overpower the force of attraction. It is a remarkable 
fact, that the members of the solar system are so adjusted to each 
other, in respect to their velocities, distances from the sun, peri- 
odic times, and gravitation towards the central body, that if any 
one of these particulars is known, all the rest become known 
also. Thus, if it were found that a new-discovered planet 
moved in its orbit six times as slow as the earth, we should know 
at once that its distance from the sun was thirty-six times as 
great as the earth's distance, that its time of revolution was two 
hundred and sixteen years, and that its gravitation towards the 
sun was twelve hundred and ninety-six times less than that of the 
earth ; for the distance is the square of the number expressing 
the rate of motion compared with that of the body taken as a 
standard ; the periodic time is the cube ; and the gravitation to 
the sun is the biquadrate of the same number. All this follows 
from Kepler's third law that the squares of the periodic times 
are as the cubes of the distances ; and from the law of univer- 
sal gravitation that the force of attraction is inversely as the 
square of the distance. The four particulars named, therefore, 
constitute a series of numbers in geometrical progression, of 
which the first term is equal to the ratio. The truth of this prop- 
osition may be demonstrated as follows. 

Let D be the mean distance of a planet from the sun, if the 
ratio of the diameter to the circumference of a circle, and P the 
time of revolution around the sun, or periodic time ; then the ex- 
pression for the velocity is V= p- x-^-. And V 2 x ^. But, 

D 2 1 

by Kepler's law, P 2 x D 3 .-. V 2 x =y 3 or V 2 x -=r. Since a body 

more remote from the sun moves more slowly in its orbit than a 
nearer body, and the comparative slowness, or retardation, is in- 
versely as the velocity, in order to avoid fractional terms, we may 
put the retardation (R) in the place of V, and then R 2 x D, (1.) 
If, therefore, R indicates how much slower a planet moves than 
another, as the earth, taken as a standard, the square of R will 
show how much farther from the sun the planet is than the 
earth. 



NUMERICAL RELATIONS. 247 

D D 3 

Again, since Vx^-, V 3 oc -p- s - But, by Kepler's law, D 3 xP 3 ; 

.. V 3 x or V 3 cz -1 and R 3 x P (2.) 

Consequently, if R expresses the retardation of a planet in 
comparison with the earth, the cube of R will express the corre- 
sponding periodic time. 

Finally, by the law of gravitation, the force of gravitation to- 
wards the central body varies as the square of the distance 

inversely, or Gx =-%> But the diminution of gravity (L) being 

inversely as the gravity, LocD 2 ; but Dx R 2 .-. D 2 x R 4 , and 
LxR 4 (3.) 

Therefore, if R denotes how much slower a planet moves in 
its orbit than the earth, R 4 will denote how much less the same 
body gravitates towards the central body. Collecting these sev- 
eral results, it appears that the square of the rate of motion gives 
the distance, its cube the periodic time, and its fourth power the 
diminution of gravity, which numbers compose a series in geomet- 
rical progression, of which the first term is the ratio. 

388. A number of very useful and convenient rules, may be 
derived from this numerical relation between the members of the 
solar system ; since, when any one of the four things named is 
given, all the rest may be found from it ; and each of the four 
may be found in four different ways when the other members of 
the series are given. This will be obvious from a few examples. 

I. Given the RATE OF MOTION or RETARDATION, (R.) 

1. Square the retardation for the distance. 

2. Cube the retardation for the periodic time. 

3. Take the fourth power of the retardation for the force of 
gravitation. 

II. Given the DISTANCE, (D.) 

1. Take the square root of the distance for the rate of motion. 

2. Take the cube of the square root of the distance for the 
periodic time. 

3. Take the square of the distance for the force of gravitation, 

III. Given the PERIODIC TIME, (P.) 



248 . THE PLANETS. 

1. Take the cube root of the periodic time for the rate of 
motion. 

2. Take the square of the cube root of the periodic time for 
the distance. 

3. Take the biquadrate of the cube root for the force of gravi- 
tation. 

IV. Given the diminished FORCE OF GRAVITATION, (L.) 

1. Take the fourth root for the rate of motion. 

2. Take the square root for the distance. 

3. Take the cube of the fourth root for the periodic time. 

V. Required the RATE OF MOTION. 

This may be obtained by taking the square root of the distance, 
or the cube root of the periodic time, or the biquadrate root of 
the force of gravitation, or by dividing the force of gravitation 
by the periodic time. 

VI. Required the DISTANCE. 

Take the square of the retardation, or the square of the cube 
root of the time, or the square root of the force of gravitation, or 
divide the time by the retardation. 

VII. Required the PERIODIC TIME. 

We may take the cube of the retardation, or the cube of the 
square root of the distance, or the cube of the fourth root of the 
gravitation, or may divide the gravitation by the retardation. 

VIII. Required the diminished GRAVITATION. 

It may be found from the fourth power of the retardation, or 
the square of the distance, or the biquadrate of the cube root of 
the time, or by multiplying the periodic time by the retardation. 

According to the foregoing rules tables may be formed, exhib- 
iting, in a striking light, the numerical relations of the members- 
of the solar system. In the following table the distances are taken 
from Herschel's Astronomy, and from these the other particulars 
are determined by the preceding rules. If Mercury were taken 
as the standard of comparison, then the retardations of all the 
other planets would be greater than unity ; but, as it is con- 
venient to take the earth as the standard, the retardations of 
Mercury and Venus will be less than unity : showing that the 
velocity (which is expressed by the fraction inverted) is greater 
than that of the earth. In like manner, the force of gravitation 



PROBLEMS. 



249 



of an inferior planet, being greater than that of the earth, is the 
reciprocal of the tabular number. 

TABLE SHOWING THE NUMERICAL RELATIONS OF THE PRIMARY 

PLANETS. 



Planets. 


Retardations. 


Distances. 


Per. Times. 


Force of Gravi- 
tation. 


Mercury, 


0.62217 


0.38710 


0.24084 


0.14985 


Venus, 


0.85049 


0.72333 


0.61519 


0.52321 


Earth, 


1.00000 


1.00000 


1.00000 


1.00000 


Mars, 


1.23440 


1.52369 


1.88080 


2.32170 


Jupiter, 


2.28100 5.20277 


11.86700 


27.06900 


Saturn, 


3.08850 


9.53878 


29.46100 


90.98900 


Uranus, 


4.37970 


19.18239 


84.01200 


367.95000 


Neptune, 


5.49040 


30.14512 


165.51000 


908.72000 



389. PROBLEMS. 

PROB. 1. The planet Pallas was discovered to have a period 
of about 4|- years. How much slower does it move in its orbit 
than the earth how much further is it from the sun and how 
much less does it gravitate towards the sun? Ans. R = 1.67, 
D=2.79, L=7.80. 

By applying the proportional numbers determined by this prob- 
lem respectively to the earth's motion per second, to its distance 
from the sun in miles, and to the space through which the earth 
departs in a second from a tangent to her orbit, we may obtain 
the numerical value of each of these elements. 

PROB. 2. What would be the periodical time of a meteor or 
planet revolving close to the earth ? 

As the moon is a body revolving around the earth at a known 
distance, and with a known periodic time, it will evidently furnish 
the necessary standard of comparison. .The distance of the moon 
from the center of the earth being 60 times the earth's radius, 
and, of course, 60 times that of the meteor, its rate of motion is 

v/60 times less. The retardation being v/60, the periodic time 

^ j$ 

will be 60 2 . Now, what part of the moon's period -is 60 2 ? Di- 
vide the moon's period (27.32 days) by 60^, and we have for the 
answer, 1 hour, 24 minutes, 38.88 seconds. 

32 



250 THE PLANETS. 

PROB. 3. What would be the periodic time of a body revolv- 
ing about the earth at the distance of 5000 miles from the cen- 
ter ? Ans. Ih. 59m. 23.28s. 

PROB. 4. How much faster must the earth revolve in order 
that bodies on its surface may lose all their gravity ? 

According to problem 2, the period of a body revolving at the 
surface of the earth, is 1.4108 hours ; and since, in a circular 
orbit, the force of gravity and the centrifugal force are equal, 
therefore a body like that contemplated in problem 2, is in equi- 
librium between these two forces ; consequently, such a body 
may be considered as having lost all its gravity, and being, by 
the supposition, close to the earth, we have only to inquire how 
much its velocity exceeds that of the earth. Now, 24 divided by 
1.4108 gives 17.01 ; which shows that were the earth to revolve 
on its axis about 17 times faster than it does at present, the bod- 
ies on the surface would lose all their weight ; and were the ve- 
locity greater than this, the centrifugal force would prevail over the 
centripetal, and the same would fly off from the earth in tangents. 

PROB. 5. Were the moon to be removed so far from the earth 
as to revolve about it but once a year, how much greater would 
be its distance than at present, how much less its velocity, and 
its gravitation towards the earth ? 

Its period being increased 13. 37 times, its retardation is 13.375" 
=2.373; its distance 2.373 2 = 5.631 ; and its diminished gravity 
5.631 2 = 31.71. Or R=2.373, D = 5.631, and L = 31.7L 

Multiplying the present distance of the moon, 238,545 miles, 
by 5.631, we obtain about 1,343,000 miles for the distance at 
which the moon must have been placed in order to complete its 
revolution in one year. 

PROB. 6. Were the earth's mass equal to the sun's, and of 
course 354,000 times as great as at present, in what time would 
the moon revolve around it ? 

Since the masses are as the cubes of the distances divided 
by the squares of the periodic times, letting the required time 
be denoted by x, I (the earth's mass) : 354.000 (the sun's mass) : : 

D 3 p\ 1 JL_ m J__ 354,000 27.32 

~WL3& : ~x z '' ' 27\32 2 ' H? " H?~~ 27.3^" " X ~ v/354,000 ~ 
6m. 7s. 



PROBLEMS. 251 

Comets, in passing their perihelion, especially when that hap- 
pens to be very near the sun, as in the great comet of 1843, move 
with an astonishing rapidity ; requiring a velocity not merely 
sufficient to generate the centrifugal force necessary to balance 
the powerful force of attraction exerted by the sun, but greatly to 
exceed that force, since they are carried far without a circular 
orbit into an elliptical or even a hyperbolic orbit. 

PROB. 7. The perihelion distance of the great comet of 1843 
being 532,000 miles from the center of the sun, what must have 
been its velocity per hour at that period ? 

PROB. 8. How much must the mass of the earth be increased 
in order that the moon may revolve about it in the same time 
as at present, when removed to three times her present dis- 
tance ? 

PROB. 9. How much must the mass of the earth be increased 
to make the moon, at her present distance, revolve jn 24 hours ? 

PROB. 10. The semi-diameter of Jupiter being 11 times that 
of the earth, and the distance of its fourth satellite from the cen- 
ter of the planet being 27 times the radius of the planet ; also the 
sidereal revolution of the satellite being 16.69 days, while that of 
the moon is 27.3217 days, and her distance 60 times the radius 
of the earth : How much does the quantity of matter in Jupiter 
exceed that of the earth ? Ans. 324.49 times. 

PROB. 11. Suppose volcanic matter to be thrown from the 
moon towards the earth, required the point where it would be in 
equilibrium between the two, the mass 'of the moon being one- 
eightieth that of the earth ? Ans. 24,000 miles from the center 
of the moon, nearly. 

PROB. 12. Suppose that the only two bodies in the universe 
were a sphere two inches in diameter, of the same density with 
the earth, for the primary, and a material point for the satellite : 
What would be the periodic time of the satellite, at the distance 
of one foot, in a circular orbit ? Ans. 2 days, 10 hours, 13 
minutes.* 



* The elements used in the solution of this problem are, for the diameter of the 
earth, 7912.4; for the distance of the moon 238,545 miles; and for its periodic 
time, 27.32 days. The solution, conducted in the ordinary mode, will be found 



CHAPTER XIII. 

COMETS METEORIC SHOWERS. 

390. A COMET,* when perfectly formed, consists of three parts, 
the Nucleus, the Envelope, and the Tail. The Nucleus, or body 
of the comet, is generally distinguished by its forming a bright 
point in the center of the head, conveying the idea of a solid, or 
at least of a very dense portion of matter. Though it is usually 

susceptible of great abridgment. But the following ingenious method is still 
shorter. It wa^ suggested to the author by one of his pupils, Mr. Samuel Emer- 
son, of the class of 1848. 

LEMMA. The periodic times of two satellites revolving about primaries of equo,l 
densities, at distances which are equimultiples of their radii, are equal. 

Demonstration. Let 

M, m, = the masses of the two bodies respectively. 

P, p = the periodic times. 

R, r = the radii of the spheres. 

D, d = ihe distances of their satellites. 

D s d* 
Then, M:m::^: ? . 

But since D and d are equimultiples of R, r, by some number n, therefore 
D 8 = RV, andd s =rV; 

Hence, M : m : : : ^ : : -^ : . But, R 3 and r 3 oc M and m. 

M m MXm MXm 
Therefore, M : m : : : -- .-. j- = ^- .-. P =p. 

The moon being distant 60.296 radii of the earth, (as would result from the above 
elements,) at the distance of 60.296 inches that of the small satellite from its pri- 
mary would be the same multiple of its radius, and, consequently, its periodic 
time the same. What then is its period at 12 inches? 

27.32* : p 9 : : 60.296 3 : 12 s .-. p = 2d. lOh. 13m. 

Corollary. If any two spheres of the same density be taken, the periodic times 
of satellites revolving about them close to the surface, will be the same in both ; 
for the case becomes this when n = \. Thus, the material point supposed in the 
above problem, will revolve about its little globe in the same time that the 
moon would revolve about the earth, both being situated close to the surfaces 
of their respective primaries. 

7, coma, from the bearded appearance of comets. 



COMETS. 253 

exceedingly small when compared with the otner parts of the 
comet, yet it sometimes subtends an angle capable of being meas- 
ured by the telescope. The Envelope (sometimes called the 
coma) is a dense nebulous covering, which frequently renders the 
edge of the nucleus so indistinct, that it is extremely difficult to 
ascertain its diameter with any degree of precision. Many com- 
ets have no nucleus, but present only a nebulous mass extremely 
attenuated on the confines, but gradually increasing in density, 
towards the center. Indeed, there is a regular gradation of com- 
ets, from such as are composed merely of a gaseous or vapory 
medium, to those which have a well-defined nucleus. In some 
instances on record, astronomers have detected with their tele- 
scopes small stars through the densest part of a comet. The 
Tail is regarded as an expansion or prolongation of the coma ; 
and presenting, as it sometimes does, a train of appalling magni- 
tude, and of a pale, portentous light, it confers on - this class oi 
bodies their peculiar celebrity. 



391. The number of comets belonging to the solar system, is 
probably very great. Many, no doubt, escape observation by 
being above the horizon in the day-time. Seneca mentions, that 
during a total eclipse of the sun, which happened 60 years before 
the Christian era, a large and splendid comet suddenly made its 
appearance, being very near the sun. . The elements of at least 
180 comets have been computed, and arranged in a catalogue for 
future comparison.* Of these, six are particularly remarkable, 
viz., the comets of 1680, 1770, and 1843; and those which bear 
the names of Halley, Encke, and Biela. The comet of 1680 was 
distinguished not only for its astonishing size and splendor, but is 
remarkable for having been the first comet whose elements were 
determined on the sure basis of mathematics, as was done by Sir 
Isaac Newton, it having appeared in his time. The comet of 
1770 is memorable for the changes its orbit has undergone by 
the action of Jupiter, and for having approached very near to the 
earth. The comet of 1843 was the most remarkable in its ap- 
pearance of all that have been seen in modern times, having been 

* See a complete catalogue of comets, whose elements have been determined, in 
the American Almanac for 1847. 



254 



COMETS. 



Fig. 76. 



Fig. 77. 





COMET OF 1811. 



COMET OF 1680. 



visible at noonday. Halley's comet (the same which reappeared 
in 1835) is distinguished as that whose return was first success- 
fully predicted, and whose orbit was first accurately determined ; 
and Biela's and Encke's comets are well known for their short 
periods of revolution, which subject them frequently to the view 
of astronomers. Biela's comet, at its return in 1846, displayed 
another remarkable feature -a separation into two distinct parts. 
This strange peculiarity was first seen from the Observatory of 
Yale College, by Messrs. Herrick and Bradley, but was first pub- 
licly announced from the Observatory at Washington. At one 
time, the distance of one nucleus from the other, was estimated 
at 157,000 miles, 

392. In magnitude and brightness, comets exhibit a great di- 
versity. They are 'sometimes so bright as to be distinctly 
visible in the day-time, even at noon and in the brightest sun- 
shine, as was the case with that of 1843 ; and such was the comet 
seen at Rome a little before the assassination of Julius Caesar 



COMETS. 255 

jji 

The comet of 1680 covered an arc of the heavens of 97, and 
its length was estimated at 123,000,000 miles.* That of 1811 
had a nucleus of only 428 miles in diameter, but a tail 132,000,000 
miles long.f Had it been coiled around the earth like a serpent, 
it would have reached round more than 5,000 times. Other com- 
ets are of exceedingly small dimensions, the nucleus being esti- 
mated at only 25 miles ; and some which are destitute of any 
perceptible nucleus, appear to the largest telescopes, even when 
nearest to us, only as a small speck of fog, or as a tuft of down. 
The majority of these bodies can be seen only by the aid of the 
telescope. 

The same comet, indeed, has often very different aspects, at its 
different returns. Halley's comet in 1305 was described by the 
historians of that age, as cometa horrendas magnitudinis ; in 1456 
its tail reached from the horizon to the zenith, and inspired such 
terror, that, by a decree of the Pope of Rome, public prayers were 
offered up at noon-day in all the Catholic churches to deprecate 
the wrath of heaven, while in 1682, its tail was only 30 in length, 
and in 1759 it was visible only to the telescope, until after it had 
passed its perihelion. At its recent return in 1835, the greatest 
length of the tail was about 12J These changes in the appear- 
ances of the same comet are partly owing to the different posi- 
tions of the earth with respect to them, being sometimes much 
nearer to them when they cross its track than at others ; also 
one spectator so situated as to see the comet at a higher angle of 
elevation or in a purer sky than another, will see the train longer 
than it appears to one less favorably situated ; but the extent 
of the changes are such as indicate also a real change in their 
magnitude and brightness. 

393. The periods of comets in their revolutions around the sun, 
are equally various. Encke's comet, which has the shortest 
known period, completes its revolution in 3^ years, or more ac- 
curately, in 1205.23 days ; while that of 1811 is estimated to have 

* Arago. f Milne's Prize Essay on Comets. 

\ But might be seen much longer by indirect vision. (Prof. Joslin, Am. Journ, 
Science, ri- 328.) 



256 COMETS. 

a period of 3383 years.* The distances to which different com- 
ets recede from the sun, are also very various. While Encke's 
comet performs its entire revolution within the orbit of Jupiter, 
Halley's comet recedes from the sun to twice the distance of 

Fig. 78. 



r 



: 



Uranus, or nearly 3600,000,000 miles. Figure 78 is a represen- 
tation, in due proportions, of the orbit of this comet. Its vast 
dimensions will be truly conceived of by reflecting that the ra- 
dius of the small dircle E of the earth's orbit implies a space of 
nearly 100,000,000 miles; that, as the comet recedes from the 
sun, it soon reaches the orbit of Jupiter, and successively traver- 
ses the orbits of Saturn, Uranus, and Neptune, reaching its aphe- 
lion 600,000,000 miles beyond the present boundaries of the 
planetary system. Some comets, indeed, are thought to go to a 
much greater distance from the sun than this, as that of 1811 
must have receded from it more than 45,000,000,000 miles, while 
some even are supposed to pass into parabolic or 'hyperbolic or- 
bits, and never to return. 

394. Comets shine by reflecting the light of the sun. In one or 
two instances they have exhibited distinct phases,^ although the 
nebulous matter with which the nucleus is surrounded, would 
commonly prevent such phases from being distinctly visible, even 
when they would otherwise be apparent. Moreover, tertain 
qualities of polarized light enable the optician to decide whether 
the light of a given body is direct or reflected ; and M. Arago, 

* Milne. f Delambre, t. 3, p. 400. 



. 

COMETS. 257 

of Paris, by experiments of this kind on the light of the comet of 
1819, ascertained it to be reflected light.* The tail of a comet 
usually increases very much as it approaches the sun ; and fre- 
quently does not reach its maximum until- after the perihelion 
passage. In receding from the sun, the tail again contracts, and 
nearly or quite disappears before the body of the comet is entirely 
out of sight. The tail is frequently divided into two portions, the 
central parts, in the direction of the axis, being less bright than 
the marginal parts. 'In 1744, a comet appeared which had six 
tails, spread out like a fan. 

The tails of comets extend in a direct line from the sun, al- 
though they are usually more or less curved, like a long quill or 
feather, being convex on the side next to the direction in which 
they are moving, (Fig. 77 ;) a figure which may result from the 
less velocity of the portions most remote from the sun. Expan- 
sions of the Envelope have also been at times observed on the 
side next the sun,f but these seldom attain any considerable 
length. 

395. The quantity of matter in comets is exceedingly small. 
Their tails consist of matter of such tenuity that the smallest stars 
are visible through them. They can only be regarded as great 
masses of thin vapor, susceptible of being penetrated through 
their whole substance by the sunbeams, and reflecting them alike 
from their interior parts and from their surfaces. It appears, 
perhaps, incredible that so thin* a substance should be visible by 
reflected light, and some astronomers have held that the matter 
of comets is self-luminous ; but it requires but very little light to 
render an object visible in the night, and a light vapor may be 
visible when illuminated throughout an immense stratum, which 
could not be seen if spread over the face of the' sky like a thin 
cloud. The highest clouds that float in our atmosphere, must be 
looked upon as dense and massive bodies, compared with the filmy 
and all but spiritual texture of a comet. J The small quantity of 



* Francoeur, 181. 

f See Dr. Joslin's remarks on Bailey's comet, Amer. Journ. Science, vol. 31. 

\ Sir J. HerscheL 

33 



258 COMETS. 

matter in comets is further proved by the fact that they have some- 
times passed very near to some of the planets without disturbing 
their motions in any appreciable degree. Thus the comet of 1770, 
in its way to the sun, got entangled among the satellites of Jupiter, 
and remained near them four months, yet it did not perceptibly 
change their motions. The same comet also came very near the 
earth ; so near, that, had its mass been equal to that of the earth, 
it would have caused the earth to revolve in an orbit so much 
larger than at present, as to have increased the length of the 
year 2h. 47m.* Yet it produced no sensible effect on the length 
of the year, and therefore its mass, as is shown by Laplace, 
could not have exceeded 5-^0 f tnat f tne earth, and might have 
been less than this to any extent. It may indeed be asked, what 
proof we have that comets have any matter, and are not mere 
reflections of light. The answer is, that, although they are not 
able by their own force of attraction to disturb the motions of 
the planets, yet they are themselves exceedingly disturbed by the 
action of the planets, and in exact conformity with the laws of 
universal gravitation. A delicate compass maybe greatly agi- 
tated by the vicinity of a mass of iron, while the iron is not sen- 
sibly affected by the attraction of the needle. 

< 

396. By approaching very near to a large planet, a comet may 
have its orbit entirely changed. This fact is strikingly exempli- 
fied in the history of the comet of 1770. At its appearance in 
1770, its orbit was found to be an ellipse, requiring for a complete 
revolution only 5| years ; and the wonder was, that it had not 
been seen before, since it was a very large and bright comet. 
Astronomers suspected that its path had been changed, and that 
it had been recently compelled to move in this short ellipse, by 
the disturbing force of Jupiter and his satellites. The French 
Institute, therefore, offered a high prize for the most complete 
investigation of the elements of this comet, taking into account 
any circumstances which could possibly have produced an altera- 
tion in its course. By tracing back its movements for some 

* Laplace. 



COMETS. 259 

years previous to 1770, it was found that, at the beginning 
of 1767, it had entered considerably within the sphere of Jupi- 
ter's attraction. Calculating the amount of this attraction from 
the known proximity of the two bodies, it was found what 
must have been its orbit previous to the time when it became sub- 
ject to the disturbing action of Jupiter. The result showed that 
it then moved in an ellipse of greater extent, having a period of 
50 years, and having its perihelion instead of its aphelion near 
Jupiter. It was therefore evident why, as long as it continued to 
circulate in an orbit so far from the center of the system, it was 
never visible from the earth. In January, 1767, Jupiter and the 
comet happened to be very near each other, and as both were 
moving in the same direction, and nearly in the same plane, they 
remained in the neighborhood of each other for several months, 
the planet being between the comet and the sun. The conse- 
quence was, that the comet's orbit was changed into a smaller 
ellipse, in which its revolution was accomplished in 5J years. 
But as it was approaching the sun in 1779, it happened again to 
fall in with Jupiter. It was in the month of June that the attrac 
tion of the planet began to have a sensible effect ; and it was not 
until the month of October following that they were finally sep- 
arated. 

At the time of their nearest approach, in August, Jupiter was 
distant from the comet only T J T of its distance from the sun, and 
exerted an attraction upon it 225 times greater than that of the 
sun. By reason of this powerful attraction, Jupiter being further 
from the sun than the comet, the latter was drawn out into a new 
orbit, which, even at its perihelion, came no nearer to the sun than 
the planet Ceres. In this third orbit, the comet requires about 
20 years to accomplish its revolution ; and being at so great a 
distance from the earth, it is invisible, and will forever remain so, 
unless, in the course of ages, it shall undergo new perturbations 
and move again in some smaller orbit as before.* 

* Milne. 



260 COMETS. 



ORBITS AND MOTIONS OF COMETS. 

397. The planets, as we have seen, (with the exception of the 
asteroids, which seem to be an intermediate class of bodies be- 
tween planets and comets,) move in orbits which are nearly 
circular, and all very near to the plane of the ecliptic, and all 
move in the same direction from west to east. But the orbits of 
comets are far more eccentric than those of the planets ; they are 
inclined to the ecliptic at various angles, being sometimes even 
nearly perpendicular to it ; and the motions of comets are some- 
times retrograde. 

398. The Elements of a comet are five, viz., (1) The perihelion 
distance ; (2) longitude of the perihelion ; (3) longitude of the node ; 
(4) inclination of the orbit; (5) time of the perihelion passage. 

The investigation of these elements is a problem extremely in- 
tricate, requiring for its solution a skilful and laborious applica- 
tion of the most refined analysis. Newton himself pronounced it 
Problema longe difficilimum ; and with all the advantages of the 
most improved state of science, the determination of a comet's 
orbit is considered one of -the most complicated problems in as- 
tronomy. This difficulty arises from several circumstances 
peculiar to comets. In the first place, from the elongated form of 
the orbits which these bodies describe, it is during only a very 
small portion of their course that they are visible from the earth, 
and the observations made in that short period cannot afterwards 
be verified on more convenient occasions ; whereas in the case 
of the planets, whose orbits are nearly circular, and whose move- 
ments may be followed uninterruptedly throughout a complete 
revolution, no such impediments to the determination of their 
orbits occur. There is also some unavoidable uncertainty in 
observations made upon bodies whose outlines are so ill-defined. 
In the second place, there are many comets which move in a 
direction opposite to the order of the signs in the zodiac, and 
sometimes nearly perpendicular to the plane of the ecliptic ; so 
that their apparent course through the heavens is rendered ex- 
tremely complicated, on account of the contrary motion of the 
earth. In the third place, as there may be a multitude of ellip- 



ORBITS AND MOTIONS OF COMETS. 261 

tic orbits, whose perihelion distances are equal, it is obvious that, 
in the case of very eccentric orbits, the slightest change in the 
position of the curve near the vertex, where alone the comet can 
be observed, must occasion a very sensible difference in the length 
of the orbit, (as will be obvious from Fig. 79;) and therefore, 
though a small error produces no perceptible discrepancy between 




the observed and the calculated course, while the comet remains 
visible from the earth, its effect, when diffused over the whole 
extent of the orbit, may acquire a most material or even a fatal 
importance. 

On account of these circumstances, it is found exceedingly dif- 
ficult to lay down the path which a comet actually follows through 
the whole system, and least of all possible to ascertain with ac- 
curacy the length of the major axis of the ellipse, and conse- 
quently the periodical revolution.* An error of only a few sec- 
onds may cause a difference of many hundred years. In this 
manner, though Bessel determined the revolution of the comet of 
1769 to be 2089 years, it was found that an error of no more 
than 5" in observation, would alter the period either to 2678 
years, or to 1692 years. Some astronomers, in calculating the 

* For when we know the length of the major axis, we can find the periodic time 
by Kepler's law, which applies as well to comets as to planets. 



262 COMETS. 

orbit of the great comet of 1680, have found the length of its 
greater axis 426 times the earth's distance from the sun, and con- 
sequently its period 8792 years ; whilst others estimate the greater 
axis 430 times the earth's distance, which alters the period to 
8916 years. Newton and Halley, however, judged that this 
comet accomplished its revolution in only 570 years. 

399. Disheartened by the difficulty of attaining to any preci- 
sion in the circumstances by which an elliptic orbit is charac- 
terized, and, moreover, taking into account the laborious calcula- 
tions necessary for its investigation, astronomers usually satisfy 
themselves with ascertaining the elements of a comet on the 
supposition of its describing a parabola; and, as this is a 'curve 
whose axis is infinite, the procedure is greatly simplified by leav- 
ing entirely out of consideration the periodic revolution. It is 
true that a parabola may not represent, with mathematical 
strictness, the course which a comet actually follows ; but as a 
parabola is the intermediate curve between the hyperbola and 
ellipse, it is found that this method, which is so much more con- 
venient for computation, also accords sufficiently with observa- 
tions, except in cases when the ellipse is a comparatively short 
one, as that of Encke's comet, for example. When the elements 
of a comet are determined, Kepler's law of areas enables astron- 
omers to find, by computation, the exact place of the comet in its 
orbit at any given time, on the supposition that its path is a pa- 
rabola ; and comparing this place with that determined by obser- 
vation for the same instant, it is seen whether the orbit is truly 
parabolic, or whether its deviation from that path is such as to 
indicate that its real path is an ellipse ; and the amount of such 
deviation will give some idea of the degree of eccentricity of the 
ellipse. 

400. The elements of a comet, with the exception of its peri- 
odic time, are calculated in a manner similar to those of the 
planets. Three good observations on the right ascension and 
declination of the comet (which are usually found by ascertaining 
its position with respect to certain stars, whose right ascensions 



ORBITS AND MOTIONS OF COMETS. 



263 



and declinations are accurately known) afford the means of cal- 
culating these elements. 

The appearances of the same comet at different periods of its 
return are so various, (Art. 392,) that we can never pronounce a 
given comet to be the same with one that has appeared before, 
from any peculiarities in its physical aspect. The identity of a 
comet with one already on record, is determined by the identity 
of the elements. When a new comet appears, we first determine 
its elements, and then turning to a catalogue of comets whose 
elements have previously been found and placed on record, we 
see whether these new elements agree with any set of those in 
the catalogue. If they do, we infer that the present comet is 
identical with that on record ; and the interval between the two 
appearances of the body will indicate its periodic time. If, for 
example, we find respecting a comet now visible in the sky, that 
its path makes the same angle with the ecliptic as that of a cer- 
tain comet in our catalogue, that it crosses the ecliptic in the same 
degree of longitude, that it comes to its perihelion in the same 
place, that its perihelion distance is the same, and its course the 
same in regard to the order of the signs, then we infer that the 
two bodies are one and the same ; and the number of years that 
have elapsed since its former appearance, indicates the period of 
its revolution around the sun. But if these particulars differ 
wholly from any set of recorded elements, we infer that the pres- 
ent is a comet which has never visited our sphere before, or at 
least one whose elements have not been determined and recorded. 
It was by this means that Halley first established the identity 
of the comet which bears his name, with one that had appeared 
at several preceding ages of the world, of which so many partic- 
ulars were left on record, as to enable him to calculate the ele- 
ments at each period. These were as in the following table. 



Time of ap- 
pearance. 


Inclination of 
the Orbit 


Longitude of 
the Node. 


Longitude of 
Perihelion. 


Perihelion 
Distance. 


Course. 


1456 


17 56' 


48 30' 


301 00 


0.58 


Retrograde. 


1531 


17 56 


49 25 


301 39 


0.57 


Retrograde. 


1607 


17 02 


50 21 


302 16 


0.58 


Retrograde. 


1682 


17 42 


50 48 


301 36 


0.58 


Retrograde. 



264 COMETS. 

On comparing these elements, no doubt could be entertained 
that they belonged to one and the same .body ; and since the in- 
terval between the successive returns was seen to be 75 or 76 
years, Halley ventured to predict that it would again return in 
1758. Accordingly, the astronomers who lived at that period, 
looked for its return with the greatest interest. It was found, 
however, that on its way towards the sun it would pass very 
near to Jupiter and Saturn, and by their action on it, would 
be retarded for a long time. Clairaut, a distinguished French 
mathematician, undertook the laborious task of estimating the 
exact amount of this retardation, and found it to be no less than 
618 days, namely, 100 days by the action of Jupiter, and 518 
days by that of Saturn. This would delay its appearance until 
early in the year 1759, and Clairaut fixed its arrival at the peri- 
helion within a month of April 13th. It came to the perihelion 
on the 12th of March. 

401. The return of Halley's comet in 1835, was looked for 
with no less interest than in 1759. Several of the most accurate 
mathematicians of the age had calculated its elements with in- 
conceivable labor. Their zeal was rewarded by the appearance 
of the expected visitant at the time and place assigned ; it tra- 
versed the northern sky, presenting the very appearances, in 
most respects, that had been anticipated ; and came to its peri- 
helion on the 16th of November, within one day of the time pre- 
dicted by Pontecoulant, a French mathematician, who had, it 
appeared, made the most successful calculation.* On its previous 
return, it was deemed an extraordinary achievement to have 
brought the prediction within a month of the actual time. 

Many circumstances conspired to render this return of Hal- 
ley's comet an astronomical event of transcendent interest. Of 
all the celestial bodies, its history was the most remarkable ; it 
afforded most triumphant evidence of the truth of the doctrine 
of universal gravitation, and consequently of the received laws of 
astronomy ; and it inspired new confidence in the power of that 

* See Professor Loomis's Observations on Halley's Comet, Amer. Journ. Science, 
209. Pontecoulant's Phys. Celeste Precis, p. 586. 



ORBITS AND MOTIONS OF COMETS. 265 

instrument (the Calculus) by means of which its elements had 
been investigated. 

402. Encke's comet, by its frequent returns, affords peculiar 
facilities for ascertaining the laws of its revolution ; and it has 
kept the appointments made for it with great exactness. On its 
return in 1839, it exhibited to the telescope a globular mass of 
nebulous matter resembling fog, and moved towards its perihelion 
with great rapidity. 

But what has made Encke's comet particularly famous, is its 
having first revealed to us the existence of a Resisting Medium 
in the planetary spaces. It has long been a question whether the 
earth and planets revolve in a perfect void, or whether a fluid of 
extreme rarity may not be diffused through space. A perfect 
vacuum was deemed most probable, because no such effects on 
the motions of the planets could be detected as indicated that 
they encountered a resisting medium. But a feather or a lock 
of cotton propelled with great velocity, might render obvious the 
resistance of a medium which would not be perceptible in the 
motions of a cannon-ball. Accordingly, Encke's comet is thought 
to have plainly suffered a retardation from encountering a resist- 
ing medium in the planetary regions. The effect of this resist- 
ance, from the first discovery of the comet to the present time, 
has been to diminish the time of its revolution about two days. 
Such a resistance, by destroying part of the projectile force, 
would cause the comet to approach nearer to the sun, and thus 
to have its periodic time shortened. The ultimate effect of this 
cause will be to bring the comet nearer to the sun at every rev- 
olution, until it finally falls into that luminary, although many 
thousand years will be required to produce this catastrophe.* It 
is conceivable, indeed, that the effects of such a resistance may 
be counteracted by the attraction of one or more of the planets 
near which it may pass in its successive returns to the sun It 
is peculiarly interesting to see a portion of matter of a tenuity 
exceeding the thinnest fog, pursuing its path in space, in obe- 



* Halley's comet, at its return in 1835, did not appear to be affected by the sup- 
posed resisting medium, and its existence is considered as still doubtful. 



266 COMETS. 

dience to the same laws as those which regulate such large and 
heavy bodies as Jupiter or Saturn. In a perfect void, a speck 
' of fog, if propelled by a suitable projectile force, would revolve 
around the sun, and hold on its way through the widest orbit, 
with as sure and steady a pace as the heaviest and largest bodies 
in the system. 

403. The most remarkable comet of the present century hith- 
erto observed, was the great comet of 1843. (See Plate I. at the 
end of the volume.) On the 28th of February of that year, the 
attention of numerous observers in various parts of the world, 
was arrested by a comet seen in the broad light of day, a little 
eastward of the sun. In Mexico it was observed, and its alti- 
tude repeatedly measured with a sextant, from nine in the morn- 
ing until sunset. In New England, it was seen at several places 
from half-past seven in the morning until after three 'in the 
afternoon, when the sky became obscured by haziness and clouds. 
Accurate measures were taken by Capt. Clark, at Portland, Maine, 
of the distance of the nucleus from the sun's limb. At 3h. 2m. 
15s. mean time, the distance of the sun's farthest limb from the 
nearest limb of the nucleus, was 4 6' 15". The comet resem- 
bled a white cloud of great density, being nearly equally shining 
throughout, with a nucleus as bright as the full moon at midnight 
in a clear sky.' During the first week in March, the appearance 
of this body was splendid and magnificent, enhanced in both 
respects by the transparency of a tropical sky, and the higher 
angle of elevation above that at which it was seen by northern 
observers. At New Haven, it was first seen after sunset, on the 
5th of March. It then lay far in the southwest. On account of 
the presence of the moon, it was not seen most favorably until 
the evening of the 17th. It then extended along the constellation 
Eridanus to the ears of the Hare, below the feet of Orion, reach- 
ing nearly to Sirius, being about 40 in length, although in the 
tropical regions its apparent length, at its maximum, was nearly 
70. It was slightly curved like a goose-quill, and colored with 
a slight tinge of rose-red, which in a few evenings disappeared, 
and left it nearly a pearly white. Our diagram (Plate I. at the 
end of the volume) presents a^pretty accurate idea of its appear- 



tfBf^b *N MOTIONS OF COMETS. 267 

ance ou the ^O^h of March, All the astronomers of the age have 
agreed in the opinion that this is one of the most remarkable ex- 
hibitions of a comet ever witnessed, although they are not fully 
agreed respecting the elements of its orbit, or its periodic time. 
Its elements resemble those of the comet of 1688, which would 
give a period of 175 years ; and to this periodic time authority 
at present inclines ;* but Prof. Hubbard, of the Washington Ob- 
servatory, after an elaborate discussion of all the observations, 
thinks the most probable period 170 years. 

Of all the comets on record, the great comet of 1843 approached 
nearest to the sun. It came within about 60,000 miles of his lumin- 
ous surface, or only about one-fourth of the distance of the moon 
from the earth. It will be recollected that to a spectator on the 
earth the sun's angular diameter but a little exceeds half a de- 
gree ; but were we to approach as near to the sun as this body 
did in its perihelion, that diameter would appear no less than 
121, 32'; and the light and heat (which increase as the square 
of the distance is diminished) would be 47,000 times as great as 
at present, the heat exceeding nearly twenty-five times that pro- 
duced by Parker's great burning lens, although this instrument 
is capable of producing effects beyond those of the most powerful 
blast furnace. The velocity of the comet was still more aston- 
ishing, being at the rate of more than one and a quarter million 
of miles per hour, a velocity sufficient to. carry it through 180, 
or half round the sun, in two hours. f 

404. Of the physical nature of comets, little is understood. It 
is usual to account for the variations which their tails undergo 
by referring them to the agencies of heat and cold. The intense 
heat to which they are subject in approaching so near the sun as 
some of them do, is alleged as a sufficient reason for the great 
expansion of the thin nebulous atmospheres forming their tails ; and 
the inconceivable cold to which they are subject in receding to 
such a distance from the sun, is supposed to account for the con- 

* See American Almanac for 1844, p. 94. Amer. Journ. of Science, xlv. 188, 
Astronomical Journal, Vol. II. p. 156. 
f Herschel's Outlines, p. 318. 



268 COMETS. 

densation of the same matter until it returns to its original dimen- 
sions. The temperature experienced by the comets of 1680 and 
1843 at their perihelion, would be sufficient to volatilize the most 
obdurate substances, and to expand the vapor to vast dimensions ; 
and the opposite effects of the extreme cold to which it would be 
subject in the regions remote from the sun, would be adequate to 
condense it into its former volume. 

This explanation, however, does not account for the direction 
of the tail, extending, as it usually does, only in a line opposite to 
the sun. Some writers therefore, as Delambre,* suppose that the 
nebulous matter of the comet, after being expanded to such a vol- 
ume that the particles are no longer attracted to the nucleus un- 
less by the slightest conceivable force, is carried off in a direc- 
tion from the sun by the impulse of the solar rays themselves. 
Others conceive of a force of repulsion, independent of any 
mechanical impulse emanating from the sun. But to assign such 
a power of communicating motion to the sun's rays while they 
have never been proved to have any momentum, or to a repulsive 
force which has no independent proof of its existence, is unphil- 
osophical ; and we are compelled to place the phenomena of 
comets' tails among the points of astronomy yet to be ex- 
pi ained.f 

405. Since those comets which have their perihelion very near 
the sun, like the comet of 1680, cross the orbits of all the planets, 
the possibility that one of them may strike the earth, has frequently 
been suggested. Still, it may quiet our apprehensions on this 

* Delambre's Astronomy, t. 3, p. 401. 

f Professor W. A. Norton, in an essay on the " Formation of Comets' Tails," main- 
tains that the head and tail of a comet do not compose one connected mass, revolving 
as one body, but that the tail is made up of particles of matter continually in the act 
of flowing away from the head, (Amer. Journal, xlvii. 104.) William Mitchell, of Nan- 
tucket, in an article, published in the 38th Yol. of the American Journal, holds that 
a comet's tail does not consist of matter at all that has the least connection with the 
comet, but is formed by the sun's rays, slightly refracted by the nucleus in traversing 
the envelope of the comet, and uniting in an infinite number of points beyond it, 
throwing a stronger than ordinary light on the etherial medium, near to or more remote 
from the comet, as the ray from its relative position and direction is more or lest 
refracted. 



METEORIC SHOWERS. 269 

subject, to reflect on the vast extent of the planetary spaces, in 
which these bodies are not crowded together as we see them 
erroneously represented in orreries and diagrams, but are sparsely 
scattered at immense distances from each other. They are like 
insects flying in the expanse of heaven. If a comet's tail lay 
with its axis in the plane of the ecliptic when it was near the 
sun, we can imagine that the tail might sweep over the earth ; 
but the tail may be situated at any angle with the ecliptic as well 
as in the same plane with it, and the chances that it will not be 
in the same plane, are almost infinite. It is also extremely im- 
probable that a comet will cross the plane of the ecliptic precisely 
at the earth's path in that plane, since it may as probably cross 
it at any other point nearer or more remote from the sun. Still, 
some comets have occasionally approached near to the earth. 
Thus Biela's comet, in returning to the sun in 1832, crossed the 
ecliptic very near to the earth's track, and had the earth been 
then at that point of its orbit, it might have passed through a 
portion of the nebulous atmosphere of the comet. The earth was 
within a month of reaching that point. This might at first view 
seem to involve some hazard ; yet we must consider that a month 
short implied a distance of nearly 50,000,000 miles. Laplace has 
assigned the consequences that would ensue in case of a direct 
collision between the earth and a comet ;* but terrible as he has 
represented them on the supposition that the nucleus of the comet 
is a solid body, yet considering a comet (as most of them doubt- 
less are) as a mass of exceedingly light nebulous matter, it is not 
probable, even were the earth to make its way directly through 
a comet, that a particle of the comet would reach the earth. The 
portions encountered by the earth, would be arrested by the at- 
mosphere, and probably inflamed ; and they would perhaps exhibit, 
on a more magnificent scale than was ever before observed, the 
phenomena of shooting stars, or meteoric showers. 

METEORIC SHOWERS. 

406. The remarkable exhibitions of shooting stars which have 
occurred within a few years past, have excited great interest 

* Syst. du Monde, L iv. c. 4. 



270 METEORIC BHOWERS. 

among astronomers, and led to some new views respecting the 
construction of the solar system. Their attention was first turned 
towards this subject by the great meteoric shower of Novembei 
13th, 1833. On that morning, from two o'clock until broad day- 
light, the sky being perfectly serene and cloudless, the whole 
heavens were lighted with a magnificent display of celestial fire 
works. At times, the air was filled with streaks of light, occa- 
sioned by fiery particles darting down so swiftly as to leave the 
impression of their light on the eye, (like a match ignited and 
whirled before the face,) and drifting to the northwest like flakes 
of snow driven by the wind ; while, at short intervals, balls of 
fire, varying in size from minute points to bodies larger than 
Jupiter and Venus, and in a few instances as large as the full 
moon, descended more slowly along the arch of the sky, often 
leaving after them long trains of light, which were, in some in- 
stances, variegated with different prismatic colors. 

On tracing back the lines of direction in which the meteors 
moved, it was found that they all appeared to radiate from the 
same point, which was situated near one of the stars (Gamma 
Leonis) of the sickle, in the constellation Leo ; and, in every rep- 
etition of the meteoric shower of November, the radiant point 
has occupied nearly the same situation. 

This shower pervaded nearly the whole of North America, 
having appeared in almost equal splendor from the British pos- 
sessions on the north, to the West India Islands and Mexico on 
the south, and from sixty-one degrees of longitude east of the 
American coast, quite to the Pacific ocean on the west. Through- 
out this immense region, the duration was nearly the same. The 
meteors began to attract attention by their unusual frequency and 
brilliancy, from nine to twelve o'clock in the evening ; were most 
striking in their appearance from two to four ; arrived at their 
maximum, in many places, about four o'clock; and continued 
until rendered invisible by the light of day. The meteors moved 
in right lines, or in such apparent curves, as, upon optical prin- 
ciples, can be resolved into right lines. Their general tendency 
was towards the northwest, although by the effect of perspective 
they appeared to move in all directions. 



METEORIC SHOWERS. 271 

407. Soon after this occurrence, it was ascertained that a sim- 
ilar meteoric shower had appeared in 1799, and what was re- 
markable, almost exactly at the same time of the year, namely, 
on the morning of the 12th of November ; and it soon appeared, 
by accounts received from different parts of the world, that this 
phenomenon had occurred on the same 13th of November, in 
1830, 1831, and 1832. Hence, this was evidently an event inde- 
pendent of the casual changes of the atmosphere ; for, having a 
periodical return, it was undoubtedly to be referred to astronom- 
ical causes, and its recurrence, at a certain definite period of the 
year, plainly indicated some relation to the revolution of the 
earth around the sun. 

It remained, however, to develop the nature of this relation, by 
investigating, if possible, the origin of the meteors. The views 
to which the author of this work was led, suggested the proba- 
bility that the same phenomenon would recur at the correspond- 
ing seasons of the year for at least several years afterwards ; and 
such proved to be the fact, although the appearances, at every 
succeeding return, were less and less striking, until 1839, when, 
so far as is known, they ceased altogether. 

Meanwhile, three other distinct periods of meteoric showers 
have also been determined ; one on the 9th of August, and (more 
rare) on the 21st of April and 7th of December respectively. 

408. The following conclusions respecting the meteoric shower 
of November, are believed to be well established, and most of 
them are now generally admitted by astronomers, though we 
cannot here exhibit the evidence on which they were founded.* 

It is considered, then, as established, that the periodical me- 
teors of November (and most of the conclusions apply equally to 
those of August) have their origin beyond the atmosphere, de- 
scending to us from some body (which, from the known consti- 
tution of the meteors, may be called a nebulous body) with which 
the earth falls in, and near or through the borders of which it 

* "We beg leave to refer the reader to various publications on the subject, by the 
author and others, in the American Journal of Science, commencing with the 25th 
volume ; and also to " Letters on Astronomy," by the author of this work. 



272 METEORIC SHOWERS. 

passes ; that this body has an independent existence as a member 
of the solar system, its periodic time being nearly commensurable 
with the earth's, either a year or half a year, so that for a number 
of years in succession the two bodies meet near the same part of 
the earth's orbit. It is further established, that the meteors con- 
sist of light combustible matter ; that they move with great 
velocities, amounting, in some instances, to not less than that of 
the earth in its orbit, or 19 miles per second ; that some of them 
are bodies of large size, sometimes several thousand feet in diam- 
eter ; that when they enter the atmosphere, they rapidly and 
powerfully condense the air before them, and thus elicit the heat 
that sets them on fire, as a. spark is elicited in the air-match, by 
being suddenly condensed by means of a piston and cylinder ; 
and that they are burned up at a considerable height above the 
earth, sometimes not less than 30 miles. 



409. Calling the body from which the meteors descended the 
" meteoric body/' it is inferred that it is a body of great extent, 
since, without apparent exhaustion, it has been able to afford such 
copious showers of meteors at so many different times ; and hence 
we regard the part that has descended to the earth only as the 
extreme portions of a body or collection of meteors, of unknown 
extent, existing in the planetary spaces. Since the earth fell in 
with the meteoric body, in the same part of its orbit for several 
years in succession, the body must either have remained there 
while the earth was performing its whole revolution around the 
sun, or it must itself have had a revolution, as well as the earth. 
No body can remain stationary within the planetary spaces ; for, 
unless attracted to some nearer body, it would be drawn directly 
towards the sun, and could not have been encountered by the 
earth again in the same part of her orbit. Nor can any mode be 
conceived in which this event could have happened so many 
times in regular succession, unless the body had a revolution of 
its own around the sun; Finally, to have come into contact 
with the earth at the same part of her orbit, in two or more suc- 
cessive years, the body must have a period which is either nearly 
the same with the earth's period, or some aliquot part of it. No 



METEORIC SHOWERS. 273 

period will fulfil these conditions, but either a year or half a year. 
Which of these is the true period of the meteoric body, is not 
fully determined. 

There are some reasons for believing that the Zodiacal Light 
(Art. 152) is the body which affords the meteoric showers, con- 
forming, as it does, to many or all of the conditions required of 
the body in question.* 



* See a paper on this subject by the author of the present work, in the Transac- 
tions of the American Association for the Advancement of Knowledge, for 1851; 
or American Journal of Science, for November, 1851. 

35 



PART III. OF THE FIXED STARS AND SYSTEM OF THE 

WORLD. 



CHAPTER I. 

OF THE FIXED STARS CONSTELLATIONS. 

410. THE FIXED STARS are s& called, because, to common 
observation, they always maintain the same situations with re- 
spect to one another. 

The stars are classed by their apparent magnitudes. The 
whole number of magnitudes recorded are sixteen, of which the 
first six only are visible to the naked eye ; the rest are telescopic 
stars. As the stars which are now grouped together under one 
of the first six magnitudes are very unequal among themselves, it 
has recently been proposed to subdivide each class into three, 
making in all eighteen instead of six magnitudes visible to the 
naked eye. These magnitudes are not determined by any very 
definite scale, but are merely ranked according to their relative 
degrees of brightness, and this is left in a great measure to the 
decision of the eye alone, although it would appear easy to meas- 
ure the comparative degree of light in a star by a photometer, 
and upon such measurement to ground a more scientific classifi- 
cation of the stars. The brightest stars to the number of 15 or 
20 are considered as stars of the first magnitude ; the 50 or 60 
next brightest, of the second magnitude ; the next 200 of the 
third magnitude ; and thus the number of each class increases 
rapidly as we descend the scale*, so that no less than fifteen or 
twenty thousand are included within the first seven magnitudes. 

411. The stars have been grouped in Constellations from the 
most remote antiquity : a few, as Orion, Bootes, and Ursa Major, 



FIXED STARS. 275 

are mentioned in the most ancient writings under the same names 
as they bear at present. The names of the constellations are 
sometimes founded on a supposed resemblance to the objects to 
which the names belong ; 'as the Swan and the Scorpion were 
evidently so denominated from their likeness to those animals ; 
but in most cases it is impossible for us to find any reason for 
designating a constellation by the figure of the animal or the hero 
which is employed to represent it. These representations were 
probably once blended with the fables of pagan mythology. The 
same figures, absurd as they appear, are still retained for the con- 
venience of reference ; since it is easy to find any particular star, 
by specifying the part of the figure to which it belongs, as when 
we say a star is in the neck of Taurus, in the knee of Hercules, 
or in the tail of the Great Bear. This method furnishes a gen- 
eral clue to its position ; but the stars belonging to any constel- 
lation are distinguished according to their apparent magnitudes, 
as follows : first, by the Greek letters, Alpha, Beta, Gamma, &c. 
Thus a Orionis, denotes the largest star in Orion, (3 Andromedce, 
the second star in Andromeda, and y Leonis, the third brightest star 
in the Lion. Where the number of the Greek letters is insuffi- 
cient to include all the stars in a constellation, recourse is had to 
the letters of the Roman alphabet, a, b, c, &c, ; and, in cases 
where these are exhausted, the final resort is to numbers. This 
is evidently necessary, since the largest constellations contain 
many hundreds or even thousands of stars. Catalogues of par- 
ticular stars have also been published by different astronomers, 
each author numbering the individual stars embraced in his list, 
according to the places they respectively occupy in the catalogue. 
These references to particular catalogues are sometimes entered 
on large celestial globes. Thus we meet with a star marked 
84 H., meaning that this is its number in Herschel's catalogue, 
or 140 M., denoting the place the star occupies in the catalogue 
of Mayer. 

412. The earliest catalogue of the stars was made by Hippar- 
chus, of the Alexandrian School, about 140 years before the 
Christian era. A new star appearing in the firmament, he was 
induced to count the stars and to record their positions, in order 



276 FIXED STARS. 

that posterity might be able to judge of the permanency of the 
constellations. His catalogue contains all that were conspicuous 
to the naked eye in the latitude of Alexandria, being 1022. Most 
persons unacquainted with the actual number of the stars which 
compose the visible firmament, would suppose it to be much 
greater than this ; but it is found that the catalogue of Hipparchus 
embraces- nearly all that can now be seen in the same latitude, 
and that on the equator, where the spectator has the northern and 
southern hemispheres both in view, the number of stars that can 
be counted does not exceed 3000. A careless view of the firma- 
ment in a clear night, gives us the impression of an infinite multi- 
tude of stars ; but when we begin to count them, they appear 
much more sparsely distributed than we supposed, and large 
portions of the sky appear almost destitute of stars. 

By the aid of the telescope, new fields of stars present them- 
selves of boundless extent ; the number continually augmenting 
as the powers of the telescope are increased. Lalande, in his 
Histoire Celeste, has registered the positions of no less than 
50,000 ; and the whole number visible in the largest telescopes 
amount to many millions. 

413. It is strongly recommended to the learner to acquaint 
himself with the leading constellations at least, and with a few of 
the most remarkable individual stars. ' The task of learning them 
is comparatively easy, when they are taken up at suitable inter- 
vals throughout the year, the moon being absent and the sky clear. 
After becoming familiar with such constellations as are visible on 
any given evening, (suppose the first of January,) these may be 
carefully reviewed after an interval of a month, and the several 
new ones added which have in the mean time risen above the 
eastern horizon. By repeating this process near the beginning 
of every month of the year, the learner will acquire a competent 
knowledge of the whole that are visible in his latitude, and with 
a small expenditure of time. It may at first be advisable to ob- 
tain, for an evening or two, the assistance of some one who is 
acquainted with the constellations, to point out such as are then 
visible in the evening sky. Then, by the aid of a celestial map, 
or, what is better, a celestial globe, the learner will pursue the 



CONSTELLATIONS. 277 

study without difficulty. We begin by rectifying the globe for 
the time, according to the directions given in Article 76. 

In the foil >wing sketch of the leading constellations, we will 
point out a few of the marks by which they may be severally 
recognized, adding occasionally a few particulars, and leaving 
it to the learner to fill up the outline by the aid of. his map or 
globe, one of which, indeed, is presumed to be before him.* 

Let us begin with the constellations of the Zodiac, which, suc- 
ceeding each other as they do in a known order, are most easily 
found, f 

ARIES (The RAM) the first constellation of the Zodiac, is known 
by two bright stars, Alpha (a) on the northeast, and Beta ((3) on 
the southwest, 4J apart, forming the head. South of Beta, at 
the distance of 2, is a smaller star, Gamma (7). The next bright- 
est star of the Ram, Delta (d), is in the tail, 15 southeast of Alpha. 
The feet of the figure rest on the head of the Whale. It has 
been already intimated, (Art. I9'3,) that the vernal equinox was 
near the head of Aries, when the signs of the Zodiac received their 
present names, but that the equinox is now found 30 westward 
of a Arietis, in consequence of the precession of the equinoxes. 

TAURUS (The BULL) will be readily found by the seven stars, 



* A celestial globe, sufficient for studying the constellations, may be purchased 
for a small sum, and is, in other respects, a valuable possession to the astronomical 
student ; but even cheap maps of the stars, like those of Burritt or Keudal, will 
answer for beginners ; and the Celestial Atlas, published by the Society for the 
Diffusion of Useful Knowledge, which is suitable for the more advanced student, 
may be procured at a moderate expense. 

f It will be expedient, where it is practicable, for the learner to study the con- 
stellations in separate portions, at different seasons of the year, as at the equinoxes 
aud at the solstices, according to the directions given in the closing article of this 
chapter. 

\ These measures are not intended to be stated with minute accuracy, but only 
with such a degree of exactness as may serve for a general guide. The learner 
will find it greatly for his advantage to accustom himself to make an accurate esti- 
mate with the eye of distances in degrees on the celestial sphere ; and he may, at 
the outset, fix on the distance between Alpha and Beta Arietis as a standard meas- 
ure (4) by which to estimate other angular distances among the stars. Thus, half 
this length applied from Beta to Gamma, indicates that the two latter stars are 2 
apart ; and two and a half times the same measure (10) will reach from the 
Pleiades to Aldebaran. Or the Pointers in the Great Bear will furnish a measure of 5 



278 FIXED STARS. 

or Pleiades, which lie in the neck, 24 eastward of a Arietis. 
The largest star in Taurus is Aldebaran, of the first magnitude, 
in the Bull's eye, 14 southeast of the Pleiades. It has a reddish 
color, and resembles the planet Mars. The other eye of the fig- 
ure is Epsilon (s), 3 northwest of Aldebaran. Five small stars, 
situated a little west of Aldebaran, in the face of the Bull, con- 
stitute the Hyades. Although the Pleiades are usually denom- 
inated the seven stars, yet it has been remarked, from a high 
antiquity, that only six are present. 

Quas septem dici, sex tamen ease solent.* Ovid. 

Some persons, however, of remarkable powers of vision, are 
still able to recognize seven, and even a greater number. f With 
a moderate telescope, not less than 50 or 60 stars, of considerable 
brightness, may be counted in this group, and a much larger num- 
ber of very small stars are revealed to the more powerful tele- 
scopes. The beautiful allusion, in the book of Job, to the " sweet 
influences of the Pleiades," and the special mention made of this 
group by Homer and Hesiod, show how early it had attracted 
the attention of mankind. The horns of the Bull are two stars, 
Beta and Zeta, situated 25 east of the Pleiades, being 8 apart. 
The northern horn, Beta, also forms one of the feet of Auriga, 
the Charioteer. 

GEMINI (The TWINS) is represented by two well-known stars, 
Castor and Pollux, in the head of the figure, 5 asunder. Castor? 
the northern, is of the first, and Pollux of the second magnitude. 
Four conspicuous stars, extending in a line from south to north, 
25 S. W. of Castor, form the feet, and two others parallel to 
these at the distance of six or seven degrees northeastward, are 
in the knees. 

CANCER (The CRAB). There are no large stars in this constel- 
lation, and it is regarded as less remarkable than any other in the 



* Their names were Electra, Maia, Taygeta, Alcyone, Celaeno, Asterope, and 
Merope, the last being the " Lost Pleiad" of the poets. Alcyone, according to a 
recent celebrated hypothesis, is distinguished as the center around which the starry 
host revolve. 

f Smyth's Cycle, II. 86. 



CONSTELLATIONS. 279 

Zodiac. The two most conspicuous stars, Alpha and Beta, are 
in the southern claws of the figure, and in its body are the 
northern and southern Asellus, which may be readily found on a 
celestial globe. But the most remarkable object in this constel- 
lation, is a misty group of very small stars, so close together 
when seen by the naked eye as to resemble a comet, but easily 
separated by the telescope into a beautiful collection of brilliant 
points. It is called Prcesepe, or the Beehive. 

LEO (The LION) is a very large constellation, and has many 
interesting members. Regulus (a Leonis) is a star of the first 
magnitude, which lies very near the ecliptic, and is much used 
in astronomical observations. North of Regulus lies a semi- 
circle of five bright stars, arranged in the form of a sickle, of 
which Regulus is the handle, and extending over the shoulder 
and neck of the Lion.* Denebola, a conspicuous star in the 
Lion's tail, lies 25 east of Regulus. Twenty bright stars in all 
help to compose this beautiful constellation. It ranges from west 
to east along the Zodiac, over more than 40 of longitude, all 
parts of the figure excepting the feet lying north of the ecliptic. 

VIRGO (The VIRGIN) extends along the Zodiac eastward from 
the Lion, covering an equally wide region of the heavens, al- 
though less distinguished by brilliant stars. Spica, however, is a 
star of the first magnitude, and lies a little east of the vernal 
equinox. Vindemiatrix, in the arm of Virgo, 18 east of Dene- 
bola, and 23 north of Spica, is easily found, and directly south 
of Denebola 13, is P Virginis ; while four other conspicuous 
stars, in the form of a trapezium, between this and Vindemiatrix, 
lie in the wing and shoulders of the figure. The feet are near 
the Balance. 

LIBRA (The BALANCE) is composed of a few scattered mem- 
bers situated between the feet of Virgo and the head of Scorpio, 
but has no very distinctive marks. Two stars of the second mag- 
nitude, Alpha on the south, and Beta 8 northeast of Alpha, 
together with a few smaller stars, form the scales. 



* As the Meteors of November always appear to radiate from a point in the bend 
of the sickle, near the star Gamma, it may be noted that the names of the six stars 
composing this figure, beginning with Regulus, are a, >?, y, 5, /*, * 



280 FIXED STARS. 

SCORPIO (The SCORPION) is one of the finest of the constella- 
tions of the Zodiac, and is manifestly so called from its resem- 
blance to the animal whose name it bears. The head is composed 
of five stars, arranged in a line slightly curved, which is crossed 
in the center by the ecliptic, nearly at right angles, a degree 
south of the brightest of the group /3 Scorpionis. Nine degrees 
southeast of this is a remarkable star of the first magnitude, 
called Antares, and sometimes the Heart of the Scorpion, (Cor 
Scorpionis.) It is of a red color, resembling the planet Mars. 
South and east of this, a succession of not less than nine bright 
stars sweep round in a semicircle, terminating in several small 
stars forming the sting of the Scorpion. The tail of the figure 
extends into the Milky Way. 

SAGITTARIUS (The ARCHER). Ten degrees eastward of the 
Scorpion's tail, on the eastern margin of the Milky Way, we come 
to the bow of Sagittarius, consisting of three stars about 6 apart, 
the middle one being the brightest, and situated in the bend of 
the bow, while a fourth star, 4 westward of it, constitutes the 
arrow. The archer is represented by the figure of a Centaur, 
(half horse and half man,) and proceeding about ten degrees east 
from the bow, we come to a collection of seven or eight stars of 
the second and third magnitudes, which lie in the human or upper 
part of the figure. 

CAPRICORNUS, (The GOAT,) represented with the head of a goat 
and the tail of a fish, comes next to Sagittarius, about 20 east- 
ward of the group that form the upper portions of that constella- 
tion. Two stars of the second magnitude, a on the north, and 
(3 on the south, 3 apart, constitute the head of Capricornus, 
while a collection of stars of the third magnitude, lying 20 south- 
east of these, form the tail. 

AQUARIUS (The WATER BEARER) is closely in contact with the 
tail of Capricornus, immediately north of which, at the distance 
of 10, is the western shoulder (/3), and 10 further east is the east- 
ern shoulder (a) of Aquarius. About 3 southeast of a is y 
Aquarii, which, together with the other two, makes an acute tri- 
angle, of which /3 forms the vertex. In the eastern arm of 
Aquarius are found four stars, which together make the figure Y, 
the open part being westward, or towards the shoulders of the 



CONSTELLATIONS. 281 

constellation. Aquarius ranges nearly 30 from north to south, 
being nearly bisected by the ecliptic. 

PISCES (The FISHES). Three figures of this kind, at a great 
distance apart, two north and one south of the ecliptic, compose 
this constellation. The Southern Fish, Piscis Australis, otherwise 
called Fomalhaut, lies directly below the feet of Aquarius, and 
being the only conspicuous star in that part of the heavens, is 
much used in astronomical measurements. It is 30 south of the 
equator. 

About 12 east of the figure Y in the arm of Aquarius, is an 
assemblage of five stars, forming a pretty regular pentagon, which 
is one of the northern members of the Constellation Pisces ; and far 
to the northeast of this figure, north of the head of Aries, lies the 
third member, the three being represented as connected together 
by a ribbon, or wavy band, composed of minute stars. 

414. The Constellations of the Zodiac being first well learned, 
so as to be readily recognized, will facilitate the learning of others 
that lie north and south of them. Let us therefore next review 
the principal Northern Constellations, beginning at the North 
Pole. 

URSA MINOR (The LITTLE BEAR). The Pole-star (Polaris) 
is in the extremity of the tail of the Little Bear. It is of the 
third magnitude, and being within less than a degree and a half 
of the North Pole of the heavens, it serves at present to indicate 
the position of the pole. It will be recollected, however, that on 
account of the precession of the equinoxes, the pole of the heav- 
ens is constantly shifting its place from east to west, revolving 
about the pole of the ecliptic, and will in time recede so far from 
the pole-star, that this will no longer retain its present distinction, 
(Art. 190.) Three stars in a straight line, 4 or 5 apart, com- 
mencing with Polaris, lead to a trapezium of four stars, the whole 
seven together forming the figure of a dipper, the trapezium being 
the body, and the three first-mentioned stars being the handle. 

URSA MAJOR (The GREAT BEAR) is one of the largest and most 
celebrated of the constellations. It is usually recognized by the 
figure of a larger and more perfect dipper than the one in the 
Little Bear three stars, as before, constituting the handle, and 

36 



282 FIXED STARS. 

four others, in the form of a trapezium, the body of the figure. 
The two western stars of the trapezium, ranging nearly with the 
North Star, are called tlie Pointers ; and beginning with the 
northern of these two, and following round from left to right 
through the whole seven, they correspond in rank to the succes- 
sion of the first seven letters of the Greek alphabet, Alpha, Beta, 
Gamma, Delta, Epsilon, Zeta, Eta. Several of them also are 
known by their Arabic names. Thus, the first in the tail, cor- 
responding to Epsilon, is Alioth, the next (Zeta) Mizar, and the 
last (Eta) Benetnasch. These are all bright and beautiful stars, 
Alpha being of the first magnitude, Beta, Gamma, Delta, of the 
second, and the three forming the tail, of the third. But it must 
be remarked that this very remarkable figure of a dipper or ladle 
composes but a small part of the entire constellation, being merely 
the hinder half of the body and the tail of the Bear. The head 
and breast of the figure, lying about ten or twelve degrees west oi 
the Pointers, contain a great number of minute stars in a trian- 
gular group. One of the fourth magnitude, Omicron, is in the 
mouth of the Bear. The feet of the figure may be looked for 
about 15 south of those already described, the 'two hinder paws 
consisting each of two stars very similar in appearance, and only 
a degree and a half apart. The two paws are distant from each 
other about 18; and following westward about the same number 
of degrees, we come to another very similar pair of stars, which 
constitute one of the fore paws, the other foot being without any 
corresponding pair. 

In a clear winter's night, when the whole constellation is above 
the pole, these various parts may be easily recognized, and the 
entire figure will be seen to resemble a large animal, readily ac- 
counting for the name given to this constellation from the ear- 
liest ages. 

DRACO (The DRAGON) is also a very large constellation, extend- 
ing for a great length from east to west. Beginning at the tail, 
which lies half way between the Pointers and the Pole-star, and 
winding round between the Great and the Little Bear, by a con- 
tinued succession of bright stars from 5 to 10 asunder, it coils 
around under the feet of the Little Bear, sweeps round the pole 
of the ecliptic, and terminates in a trapezium formed by four con- 



CONSTELLATIONS. 283 

spicuous stars, from thirty to thirty-five degrees from the North 
Pole. A few of the members of this constellation are of the sec- 
ond, but the greater part of the third magnitude, and below it. 

* 

415. With the constellations already described as general land- 
marks, we may now proceed with each of the principal remaining 
ones, by stating its boundaries, as we do those of countries in 
geography ; their relative situations being thus first learned from 
a map, or (what is better) from a celestial globe, and then being 
severally traced out on the sky itself. We will begin with those 
which surround the North Pole. 

CEPHEUS (The KING) is bounded N. by the Little Bear, E. by 
Cassiopeia, S. by the Lizard, and W. by the Dragon. The head 
lies in the Milky Way, and the feet extend towards the pole. 
It contains no stars above the third magnitude. 

CASSIOPEIA is bounded N. and W. by Cepheus, E. by Camel- 
opardalus, and S. by Andromeda, and is one of the Constellations 
of the Milky Way. It is readily distinguished by the figure of a 
chair inverted, of which two stars constitute the back, and four, 
in the form of a square, the body of the chair. It is on the op- 
posite side of the pole from the Great Bear, and nearly at the 
same distance from it. 

CAMELOPARDALUS (The GIRAFFE) is bounded N. by the Little 
Bear, E. by the head of the Great Bear, S. by Auriga and Per- 
seus, and W. by Cassiopeia. Although this Constellation occu- 
pies a large space, yet it has no conspicuous stars. 

ANDROMEDA is bounded N. by Cassiopeia, E. by Perseus, S. by 
Pegasus, and W. by the lizard. The direction of the figure is 
from S. W. to N. E., the head coming down within 30 of the 
equator, and being recognized by a star of the second magnitude, 
which forms the northeastern corner of the great square in Pega- 
sus, to be described hereafter. At the distance of six or seven 
degrees from the head, are three conspicuous stars in a row, 
ranging from north to south, which lie in the breast of the figure ; 
and about the same distance from these, and parallel to them, 
three more, which constitute the girdle of Andromeda. Near the 
northernmost of the three, is a faint, misty object, often mistaken 



284 FIXED STARS. 

for a comet, but is a nebula, and one of the most remarkable in 
the heavens. 

PERSEUS is bounded N. by Cassiopeia, E. by Auriga, S. by 
Taurus, and W. by Andromeda. The figure extends from north 
to south, and is represented by a giant holding aloft a sword in 
his right hand, while his left grasps the head of Medusa, a group 
of stars on the western side of the figure, embracing the celebra- 
ted star Algol. A series of bright stars descend along the shoul- 
ders and the waist, and there divide into the two legs. The 
western foot is 8 north of the Pleiades. The eastern leg is bent 
at the knee, which is distinguished by a group of small stars. 
Near the sword handle, under Cassiopeia's 'chair, is a fine cluster 
of stars, so close together as scarcely to be separable by the eye. 

AURIGA (The WAGONER) is bounded N. by Camelopardalus, 
E. by the Lynx, S. by Taurus, and W. by Perseus. He is rep- 
resented as bearing on his left shoulder the little Goat Capella, a 
white and beautiful star of the first magnitude, (a Aurigse,) while 
Beta forms the right shoulder, 8 east of Capella. These two 
bright stars form, with the northern horn of the Bull, at the dis- 
tance of 18, an isosceles triangle. 

LEO MINOR (The LESSER LION) is bounded N. by Ursa Major, 
E. by Coma Berenices, S. by Leo, and W. by the Lynx. It lies 
directly under the hind feet of the Great Bear, and over the sickle 
in Leo, and is easily distinguished. Four stars in the central 
part of the figure, from 4 to 5 apart, form a pretty regular par- 
allelogram. 

CANES VENATICI (The GREYHOUNDS). This Constellation lies 
between the hind legs of the Great Bear on the west, and Bootes 
on the east ; Cor Caroli, a solitary star of the third magnitude, 
18 south of Alioth, in the tail of the Great Bear, will serve to 
mark this Constellation. 

COMA BERENICES (BERENICE'S HAIR) is a cluster of small stars, 
composing a rich group, 15 N. E. of Denebola, in the Lion's 
tail, in a line between this star and Cor Caroli, and half way be- 
tween the two. 

BOOTES is bounded N. by Draco, E. by the Crown and the 
head of Serpentarius, S. by Virgo, and W. by Coma Berenices and 
the Hounds. It reaches for a great distance from north to south, 



CONSTELLATIONS. 285 

tne head being within 20 of the Dragon, and the feet reaching 
to the Zodiac. In the knee of Bootes is Arcturus, a star of the 
first magnitude. The next brightest star, Beta, is in the head of 
Bootes, 23 north of Arcturus, and 15 east of the last star in the 
tail of the Great Bear. 

CORONA BOREALIS (The NORTHERN CROWN) is bounded N. and 
E. by Hercules, S. by the head of Serpentarius, and W. by 
Bootes. It is formed of a semicircle of bright stars, six in num- 
ber, of which Gemma, near the center of the curve, is of the sec- 
ond magnitude. 

HERCULES is bounded N. by Draco, E. by Lyra, S. by Ophiu- 
chus, and W. by Corona Borealis. It is a very large Constellation, 
and contains some brilliant objects for the telescope, although its 
components are generally very small. The figure lies north and 
south, with the head near the head of Ophiuchus, and the feet un- 
der the head of Draco. Being between the Crown and the Lyre, 
its locality is easily determined. The eastern foot of Hercules 
forms an isosceles triangle with the two southern stars of the tra- 
pezium in the head of Draco ; while the head of Hercules is far 
in the south, within 15 of the equator, being 6 west of a similar 
star which constitutes the head of Ophiuchus. 

LYRA (The LYRE) is bounded N. by the head of Draco, E. by 
the Swan, S. and W. by Hercules. Alpha Lyrse, or Vega, is of 
the first magnitude. It is accompanied by a small acute triangle 
of stars. Its color is a shining white, resembling Capella and the 
Eagle. 

CYGNUS (The SWAN) extends along the Milky Way, below 
Cepheus, and immediately eastward of the Lyre, and has the fig- 
ure of a large bird flying along the Milky Way from north to 
south, with outstretched wings and long neck. Commencing 
with the tail, 25 east of Lyra, and following down the Milky 
Way, we pass along a line of conspicuous stars which form the 
body and neck of the figure ; and then returning to the second of 
the series, we see two bright stars at eight or nine degrees on the 
right and left (the three together ranging across the Milky Way) 
which form the wings of the Swan. This Constellation is among 
the few, which exhibit some resemblance to the animals wnose 
names they bear. 



286 



FIXED STARS. 



VULPECULA (The LITTLE Fox) is a small Constellation,-in which 
a fox is represented as holding a goose in his mouth. It lies in 
the Milky Way, between the Swan on the north and the Dolphin 
and the Arrow on the south. 

AauiLA (The EAGLE) stretches across the Milky Way, and is 
bounded N. by Sagitta, a small Constellation which separates it 
from the Fox, E. by the Dolphin, S. by Antinous, and W. by 
Taurus Poniatowski, (the Polish Bull,) which separates it from 
Ophiuchiis. It is distinguished by three bright stars in' the neck, 
known as the " three stars," which lie in a straight line about 2 
apart, on the eastern margin of the Milky Way. The central 
star is of the first magnitude. Its Arabic name is Altair. 

ANTINOUS lies across the equator, between the Eagle on the 
north, and the head of Capricorn on the south. 

DELPHINUS (The DOLPHIN) is situated east and north of Altair, 
and is composed of five stars of the third magnitude, of which 
four, in the form of a rhombus, compose the head, and the fifth 
forms the tail. 

PEGASUS (The FLYING HORSE) is a very large Constellation, and 
is bounded N. by the Lizard and Andromeda, E. and S. by Pisces, 
W. by the Dolphin. The head is near the Dolphin, while the 
back rests on Pisces, and the feet extend towards Andromeda. 

A large square, composed of four conspicuous members, one 
(Markab) of the first, and three others of the second magnitude, 
distinguish this Constellation. The corners of the square are 
abput 15 apart; the northeastern corner being in the head of 
Andromeda. 

OPHIUCHUS is another very large Constellation, the head being 
near the head of Hercules, and the feet reaching to Scorpio, the 
western foot being almost in contact with Antares. The figure 
is that of a giant holding a serpent in his hands. The head of 
the serpent is a little south of the Crown, and the tail reaches 
far eastward towards the Eagle. 

416. Of the Constellations which lie south of the Zodiac, we shall 
notice only Cetus, Orion, Lepus, Monoceros, Canis Major, Canis 
Minor, Hydra, Crater, and Corvus. 

CETUS (The WHALE) is distinguished rather for its extent than 



CONSTELLATIONS. 287 

its brilliancy, occupying a large tract of the sky south of the Con- 
stellations Pisces and Aries. The head is directly below the head 
of Aries, and the tail reaches westward 45, being about 10 south 
of the vernal equinox. Menkar, (a Ceti,) the largest of its com- 
ponents, is situated in the mouth, 25 southeast of a Arietis ; and 
Mira (o Ceti) in the neck, 14 west of Menkar, is celebrated as a 
variable star, which exhibits different magnitudes at different 
times. 

ORION is one of the most magnificent of the Constellations, and 
one of those that have longest attracted the admiration of man- 
kind, being alluded to in the book of Job, and mentioned by Ho- 
mer. The head of Orion lies southeast of Taurus, 15 from 
Aldebaran, and is composed of a cluster of small stars. Two very 
bright stars, Betalgeuse of the first, and Bellatrix of the second 
magnitude, form the shoulders ; three more, resembling the three 
stars of the Eagle, compose the girdle ; and three smaller stars, 
in a line inclined to the girdle, form the sword. Rigel, of the first 
magnitude, makes the west foot, but the corresponding star, 9 
southeast of this, which is sometimes taken for the other foot, is 
above the knee, this foot being concealed behind the Hare. 
Orion's club is marked by three stars of the fifth magnitude, close 
together, in the Milky Way, just below the southern horn of the 
Bull. Orion is a favorite Constellation with the practical astron- 
omer, abounding, as it does, in addition to the splendor of its 
components, with fine nebula, double stars, and other objects of 
peculiar interest when viewed with the telescope. It embraces 
70 stars, plainly visible to the naked eye, including two of the 
first, four of the second, and three of the third magnitude. 

LEPUS (The HARE). Below Rigel, the western foot of Orion, 
is a small trapezium of stars, which forms the ears of the Hare ; 
and an assemblage of nine stars, of the third and fourth magni- 
tudes, south and east of these, make up the remaining parts of the 
figure. 

CANIS MAJOR (The GREATER DOG) lies directly east of the 
Hare, and is highly distinguished by containing Sirius, the most 
splendid of all the fixed stars, which lies in the mouth of the fig- 
ure. In the fore paw, 6 west of Sirius, is a star of the second 
magnitude, (,3 Canis Majoris,) and from 10 to 15 south of Sir- 



288 FIXED STARS. 

ius, is a collection of stars of the second and third magnitudes, 
which make up the hinder portions of the figure. The Egyptians, 
who anticipated the rising of the Nile by the appearance of Sir- 
ius in the morning sky, represented the Constellation by the figure 
of a dog, the symbol of a faithful watchman. 

CANIS MINOR (The LESSER DOG). About 25 north of Sirius, 
is the bright star Procyon, also of the first magnitude, which 
marks the side of the Lesser Dog. A star of the third magnitude , 
(/3), 4 northwest of this, in the head of the figure, forms with 
Procyon the lower side of an elongated parallelogram, of which 
Castor and Pollux, 25 north, form the upper side. 

MONOCEROS is a large Constellation, occupying the space be- 
tween the Greater and the Lesser Dog, but has. no conspicuous 
members. 

HYDRA occupies a long space south of Leo, Virgo, and Libra. 
Its head, which is south of the fore paws of the Lion, consists of 
four stars of the fourth magnitude, of nearly uniform appearance ; 
and about 15 S. E. of these is the Heart, (Cor Hy dree,) 23 south 
of Regulus. Resting on Hydra, and south of the hind feet of 
Leo, is Crater, (the Cup,) consisting of six stars of the fourth 
magnitude, arranged in the form of a semicircle ; and a little 
further east, also perched on the back of Hydra, is Corvus, (the 
Crow,) the two brightest components of which are situated in one 
of the wings of the figure, in a line between Crater and Spica 
Virginis. 

417. According to an intimation given in a note on p. 277, the 
Constellations may be advantageously studied at four different 
periods of the year, as near the equinoxes and the solstices, accord- 
ing to the following directions. The latitude supposed is 41. 

LESSON I. For the middle of September, from 8 to 10 o'clock. 
At 8 o'clock Scorpio is near setting in the S. W., Antares being 
10 high. The bow of Sagittarius is seen on the eastern margin 
of the Milky Way, the arrow being directed to a point a little 
below Antares. At 9 o'clock, the horns of the Goat come upon 
the meridian ; and at 10 o'clock, the western shoulder of Aqua- 
rius. The other shoulder, and the figure Y in the arm, may also 



CONSTELLATIONS. 289 

be easily found from the description given on p, 280 ; also, the 
Pentagon, in Pisces, and Fomalhaut, (the Southern Fish,) a soli- 
tary bright star far in the south, only 16 above the horizon. 
The head of Aries appears in the east, and the Pleiades are but 
little above the horizon, while Aldebaran is just rising. Return- 
irg now to the west, (at 10 o'clock,) the Crown is seen a little 
north of west, about 20 high ; Lyra is 30 west of the zenith ; the 
Swan is nearly overhead : and following down the Milky Way, 
the Eagle is seen on its eastern margin over against Lyra on the 
western ; and the Dolphin, a little eastward of the Eagle, and as 
far above the horns of Capricornus, as the latter are above the 
southern horizon. Following on east of the meridian, the great 
square in Pegasus may next be identified ; and since the north- 
eastern corner of the square is in the head of Andromeda, this 
Constellation may next be learned ; and then Perseus and Auri- 
ga, which appear still further east. Directly north of Perseus, is 
Cassiopeia's chair ; and next to that we may take the Pole Star, 
the Little Bear, and the Great Bear, the Dipper only being 
traced for the present. Commencing now at the tail of the 
Dragon, we may trace round this figure between the two Bears 
to the head, which brings us back to Lyra and the head of Her- 
cules. The boundaries of this Constellation, and of Ophiuchus, 
which lies south of it, will end the first lesson. 

LESSON II. For the middle of December, from 7 to 10 o'clock. 
Of the Constellations of the Zodiac, Taurus and Gemini are now 
favorably situated for observation in the east. At 7 o'clock, the 
tail of Cetus just reaches the meridian, its head being seen below 
the feet of Aries. Orion is just risen in the S. E. At 9 o'clock, 
just above the western horizon, are seen in succession from south 
to north, Aquarius, the Dolphin, the Eagle, the Lyre, and the 
Dragon's head. Between the Eagle and the Lyre, at a little 
higher altitude, we perceive the Swan, flying directly downwards. 
Between the tail of the Swan and the Pole Star, is Cepheus ; 
and from the pole, along the meridian, we trace Cassiopeia, the 
feet of Andromeda, the head of Aries, and the neck of the Whale. 
At 10 o'clock, Perseus has reached the meridian, the star Algol, 
in the head of Medusa, being directly over head. The Pleiades 

37 



290 FIXED STARS. 

are but little eastward of the zenith ; and following along south 
from the pole, at the interval of from one to two hours east of the 
meridian, we may trace in succession, Camelopard, Auriga, Tau- 
rus, Orion, and the Hare. Turning along the eastern horizon, 
we find Canis Major, Monoceros, Canis Minor, the head of Hy- 
dra, (just rising,) Cancer, Leo, the sickle just appearing about 3 
north of the east point. Leo Minor and Ursa Major complete the 
survey ; and we may now advantageously trace ou-t the various 
parts of the Great Bear, as described on p. 281 ; the two stars 
composing its hindmost paw being scarcely above the horizon. 

j 

LESSON III. For the middle of March, from 8 to 10 o'clock. 
At 8 o'clock, we see the Twins nearly overhead, and Procyon 
and Sirius, at different intervals, towards the south. Along the 
west we recognize the neck and head of the Whale, the head of 
Aries, and the head of Andromeda ; next above these, Orion, 
Taurus, Perseus, Cassiopeia, and Cepheus; and north of the head 
of Orion, we see Auriga and Camelopard. In the S. W., Hydra 
is now fully displayed ; and following on north, we obtain fine 
views of the Greater and the Lesser Lion, and the Great Bear. 
At 9 o'clock, Crater and Corvus appear in the S. E. on the back 
of Hydra ; Virgo extends from Leo down to the horizon, Spica 
Virginis being about 5 high ; and north of Virgo, we trace in 
succession Coma Berenices, Cor Caroli, Bootes, with Arcturus, 
and the Crown lying far in the N. E. 

LESSON IV. For the middle of June, from 9 to 10 o'clock. 
At 9 o'clock, Bootes, Corona Borealis, the head of Libra, the Ser- 
pent, and Scorpio, lie along on either side of the meridian. Castor 
and Pollux are just setting, and Leo is about an hour high. East 
of Leo, Virgo is seen extending along towards the meridian, Spica 
being about 30 above the southern horizon. North of Leo and 
Virgo, we recognize Leo Minor, Coma Berenices, Cor Caroli, 
and Ursa Major. At 10 o'clock ; we trace along the eastern side 
of the meridian, Draco, Hercules, and Ophiuchus ; and east of 
these, the Lyre, the Eagle, Antinous, Sagittarius, and Capricor- 
nus. N^rth of the Eagle, and round to the east, we find Cepheus 



DOUBLE STARS. 291 

and Cassiopeia, Andromeda rising in the northeast, Pegasus in 
the east, and Aquarius in the southeast. Thus we may advan- 
tageously complete a review of the Constellations. 



CHAPTER II. 

DOUBLE STARS TEMPORARY STARS VARIABLE STARS CLUSTERS 

AND NEBULAE. 

418. THE view hitherto taken of the starry heavens presents 
little that is new, since most of the Constellations, visible in our 
latitude, and the most conspicuous of the individual stars, have 
been known from antiquity. But the objects to be described in 
the present chapter, are chiefly such as have been discovered by 
modern astronomy, aided by the powerful telescopes which, since 
the time of Sir William Herschel, have been directed to the heav- 
ens. Different orders and systems of stars have been brought to 
light, and a new and still more wonderful class of bodies, called 
Nebulae, have been reached in the depths of the stellar universe. 

419. The introduction into practical astronomy of Herschel's 
great Forty Feet Reflector, in 1789, was a great event in the 
study of the stars. This instrument, in its previous humble 
forms, had been very little employed upon the stars, they being 
supposed to be too remote for its powers, which seemed only 
suited to nearer worlds, as the sun and planets. It was not, how- 
ever, ..an increase of magnifying power that was wanted for 
researches on these distant objects, but an increase of light, by 
which a few scattered rays sent to us from bodies hidden in the 
depths of space, might be collected in such numbers, and directed 
into the eye, as would render visible objects otherwise invisible, 
not because they do not transmit to us any light, but because not 
enough of what they transmit enters the small pupil of the eye 
for the purposes of distinct vision. Telescopes of great aperture, 
therefore, by collecting a large beam of light and * t nveying it to 



292 FIXED STARS. 

the eye, greatly enlarge the powers of this organ, and enable it to 
penetrate proportionally further into the most distant regions of 
the universe. Sir W. Herschel himself made wonderful progress 
in the knowledge of the starry heavens, and by his own researches 
discovered a large portion of those bodies which we are now to 
describe ; and his son, Sir John Herschel, has cultivated, with 
great success, the same field, and especially, by a residence of 
five years at the Cape of Good Hope, devoted assiduously to ob- 
servations with large instruments, has greatly augmented our 
knowledge of the stellar systems of the southern hemisphere. 
Moreover, telescopes of still greater power than that of the elder 
Herschel, and especially instruments capable of nicer angular 
measurements, have recently enriched the department of practical 
astronomy. The most remarkable of these are the grand JKe- 
flector constructed by Lord Rosse, an Irish nobleman, and 
the great Refractors belonging respectively to the Pulkova and 
Cambridge Observatories. Lord Rosse 's telescope considerably 
exceeds in dimensions and in power the forty feet reflector of Sir 
W. Herschel, being 50 feet in focal length, and having a diame- 
ter of 6 feet, whereas that of the Herschelian telescope was only 
4 feet. This unexampled magnitude makes this instrument su- 
perior to all others in light, and fits it pre-eminently for observa- 
tions on the most remote and obscure celestial objects, such as 
the faintest nebulae. But its unwieldy size, and its liability to 
loss of power, by the tarnishing or temporary blurring of the great 
speculum, will render it far less available for actual research than 
the great refractors which come in competition with it. Until 
recently, it was thought impossible to form perfect achromatic 
object-glasses of more than about five inches diameter ; but they 
have been successively enlarged, until we can no longer set 
bounds to the dimensions which they may finally assume. The 
Pulkova telescope (at St. Petersburgh) has a clear aperture of about 
15 inches, and a focal length of 22 feet. The telescope recently 
acquired by Harvard University, is perhaps the finest refractor 
hitherto constructed. It was made by the same artists, and upon 
the same scale with that, but its performances are thought even 
to exceed those of the Pulkova instrument. We now proceed to 
review some oi the discoveries among the stars, which the re- 



DOUBLE STARS. 293 

searches made with such instruments as the foregoing have 
brought to light. 

DOUBLE STARS. 

420. DOUBLE STARS are those which appear single to the naked 
eye, but are resolved into two by the telescope ; or if not visible 
to the naked eye, are seen in the telescope very close together. 
Sometimes three or more stars are found in this near connection, 
constituting triple or multiple stars.* Castor, for example, when 
seen by the naked eye, appears as a single star ; but in a tele- 
scope, even of moderate powers, it is resolved into two stars, be- 
tween the third and fourth magnitudes, within 5" of each other. 
These two stars are of nearly equal size, but frequently one is 
exceedingly small in comparison with the other, resembling a sat- 
ellite near its primary, although in distance, in light, and in other 
characteristics, each has all the attributes of a star, and the com- 
bination, therefore, cannot be that of a planet with a satellite. 
The distance between these objects varies from a fraction of a 
second to thirty-two seconds. In some cases, the extreme close- 
ness, and the exceeding minuteness of double stars, require, for 
their separation, the best telescope, united with the most acute 
powers of observation. Indeed, certain of these objects are re- 
garded as the severest tests both of the excellence of the instru- 
ment, and of the skill of the observer. 

421. When Sir William Herschel began his observations on 
double stars, about the year 1780, he was acquainted with only 4. 
By his own researches he extended the number to 2400. Sir 
John Herschel, Sir James South, and M. Struve, the great Russian 
astronomer, prosecuted the same line of research ; and when 
Sir John Herschel left England for the Cape of Good Hope, in 
1833, the whole number of double stars enrolled was 3346 ; and 
this number was increased, by that eminent astronomer, by adding 
those of the southern hemisphere, to 5542. It appears, therefore, 
that the number of double stars considerably exceeds all the stars 



* See several figures of double and multiple stars, in Plate III. at the end of 
the volume. 



294 FIXED STARS. 

visible to the naked eye. In some instances, this proximity arises 
undoubtedly from the two members lying nearly in the same line 
of vision, and therefore being projected very near to each other 
on the face of the sky ; but in most cases the double stars are 
proved to have a physical relation to each other, and are therefore 
said to be physically double, while the former are said to be opti- 
cally double. There is no longer any doubt that among the stars 
are separate systems^ in which two, three, and even in one in- , 
stance at least, six stars are bound together in relations of mutual 
dependance, suns with suns, as the members of the solar system 
compose an individual province in the great empire of nature. A 
star in Orion's sword (Theta Orionis) has been for some time 
known as a quadruple star, the members of which form a small 
trapezium; and recent observations have detected in two of 
these, severally, companions of extreme minuteness, the whole 
composing a figure like the following : 



Many of the double stars are distinguished by the components 
exhibiting different colors, often finely contrasted with each other ; 
as orange with blue or green, yellow with blue, and white with 
purple. Gamma Andromedse is a close double star, the compo- 
nents of which are both green. Insulated stars of a red color, 
almost as deep as that of blood, occur in many parts of the heav- 
ens, but no green or blue star of any decided hue has ever been 
noticed unassociated with a companion brighter than itself.* 



422. TEMPORARY STARS. 

TEMPORARY STARS are new stars which have suddenly made 
their appearance, and after a certain interval, as suddenly disap- 
peared, and returned no more. It was the appearance of a new 
star of 'this kind, 125 years before the Christian era, that prompted 

* HerscheL 



VARIABLE STARS. 295 

Hipparchus to form a catalogue of the stars, the first on record. 
Such also was the star which suddenly shone out, A. D. 389, in 
the Eagle, as bright as Venus, and after remaining three weeks, 
disappeared entirely. At other periods, at distant intervals, sim- 
ilar phenomena have presented themselves. Thus the appear- 
ance of a new star in 1572 was so sudden, that Tycho Brahe, 
returning home one evening, was surprised to find a collection of 
country people gazing at a star, which he was sure did not exist 
half an hour before. It was then as bright as Sirius, and continued 
to increase until it surpassed Jupiter when "brightest, and was visi- 
ble at midday. In a month it began to diminish, and in three 
months afterwards it had entirely disappeared. Some stars are 
now missing which were registered in the older catalogues. In 
one instance, at least, (that of Neptune,) the supposed star has 
proved to have been a planet. 

423. VARIABLE STARS. 

VARIABLE STARS are those which undergo a periodical change 
ef brightness. One of the most remarkable is the star Mira, in 
the neck of the Whale (Omicron Ceti). It appears once in 11 
months, remains at its greatest brightness about a fortnight, being 
then, on some occasions, equal to a star of the second magnitude. 
It then decreases about three months, until it becomes completely 
invisible, and remains so about five months, when it again be- 
comes visible, and continues increasing during the remaining 
three months of its period. 

Another very remarkable variable star is Algol (f3 Persei). It 
is suddenly visible as a star of the second magnitude, and con- 
tinues such for 2d. 14h., when it begins rapidly to diminish in 
splendor, and in about 3J hours is reduced to the fourth magni- 
tude. It then begins again to increase, and in 3J hours more, is 
restored to its usual brightness, going through all its changes in 
less than three days. This remarkable law of variation appears 
strongly to suggest the revolution round it of some opake body, 
which, when interposed between us and Algol, cuts off a large 
portion of its light. It is (says Sir J. Herschel) an indication of 
a high degree of activity in regions where, but for such evidence, 



296 FIXED STARS. 

we might conclude all to be lifeless. Our sun requires almost nine 
times this period to perform a revolution on its axis. On the 
other hand, the periodic time of an opake revolving body, suffi- 
ciently large, which would produce a similar temporary obscura- 
tion of the sun, seen from a fixed^star, would be less than fourteen 
hours. 

The duration of these periods is e'xtremely various. While that 
of (3 Persei, above mentioned, is less than three days, others are 
more than a year, and others many years. 

424. CLUSTERS AND NEBULAE. 

In various parts of the firmament are seen large groups, or 
CLUSTERS, which, either by the naked eye, or by the aid of the 
smallest telescope, are perceived to consist of a great number of 
small stars. Such are the Pleiades, Coma Berenices, and Pras- 
sepe, or the Bee-hive, in Cancer. The Pleiades, or Seven Stars, 
as they are called, in the neck of Taurus, is the most conspicuous 
cluster. When we look directly at this group,, we cannot distin- 
guish more than six stars, but by turning the eye sideways* upon 
it, we discover that there are many more. The telescope only 
can, however, display the real magnificence of the Pleiades. (See 
Plate III. Fig. 1.) Coma Berenices has fewer stars, but they are 
of a larger class than those which compose the Pleiades. The 
Bee-hive, or Nebula of Cancer, is one of the finest objects of this 
kind for a small telescope, being, by its aid, converted into a rich 
congeries of shining points. A cluster in the sword-handle of 
Perseus, below Cassiopeia's chair, though but a dim speck to the 
naked eye, is a very elegant object to a large telescope, being 
separated into bright and beautiful stars, embracing several dis- 
tinct subordinate clusters of exceedingly minute stellar points. 
The head of Orion affords an example of another cluster, though 
less remarkable than the others. - , 



* Indirect vision is far more delicate than direct. Thus we can see the Zodiacal 
Light or a comet's tail much more distinctly and better defined, (partly, perhaps, 
by the effect of contrast,) if we fix one eye on a part of the sky at some distance, 
and turn the other eye obliquely upon the object. 



CLUSTERS AND NEBULAS. 297 

425. NEBULAE are faint misty objects seen in various parts of the 
firmament, always maintaining a fixed position, which resemble 
comets, or a speck of fog. The Galaxy, or Milky Way, presents 
a constant succession of large nebula. Of the individual nebu- 
las, seen by the naked eye, the most conspicuous is that near the 
girdle of Andromeda. It is the oldest known nebula, having at- 
tracted the attention of star-gazers as early as the beginning of 
the tenth century,* although it is commonly said to have been 
discovered by Simon Marius, in 1612. No powers of the tele- 
scope have been able to resolve this into separate stars, although 
the great Cambridge telescope reveals a vast number of stars, 
more than 1500, of various degrees of brightness, scattered over 
its surface ; but these appear not to belong to the nebula itself, 
which has hitherto afforded no evidence of resolution, f Its 
dimensions are astonishingly great, since it covers a space of a 
quarter of a degree in diameter ; and we must bear in mind that, 
at such a distance as the fixed stars, a space of 15' implies an im- 
mense extent. Its figure is oval, and elliptical nebulas constitute 
a common variety among the figures which these bodies exhibit. 
(See Plate III. Fig. 2, for a representation of the great nebula of 
Andromeda.) Another very common figure are the globular 
nebulas. A grand specimen of this variety may be easily found 
in the Constellation Hercules, between Zeta and Eta. Draw a 
line from Lyra to Gemma of the Crown, and 3 above the center 
of that line will be the place of this nebula. When viewed with 
a small telescope, it exhibits only a globular cloud, (Plate III. 
Fig. 3, a,) but to a more powerful instrument it reveals its real 
glories in a form truly exciting to the beholder, (Fig. 3, b.) About 
4000 nebulas have been detected and described, of which about 
1700 have recently been added by Sir John Herschel, from his Re- 
sults of Observations at the Cape of Good Hope. Among the latter 
are two remarkable spots, well known to navigators, situated near 
the south pole, called Magellanic Clouds by sailors, but by as- 
tronomers, the Nubecula Major and the Nubecula Minor. They 
are found to consist of a wonderful collection of nebulas, the 



* Smyth's Cycle, II. 15. 
f Memoirs of the Amer. Acad. VoL IIL 
38 



FIXED STARS 

greater .embracing 278 nebulae, and the lesser 37. Both together 
compose a most magnificent assemblage. In the sword of Orion 
is a celebrated nebula, long known, which, until recently, had 
resisted all attempts to resolve it into stars ; but the great Reflec- 
tor of Lord Rosse, and more recently the great Refractor of the 
Cambridge Observatory, have succeeded in a partial resolution, 
at least, of this grand object, and have authorized the anticipation 
that, with a small increase of telescopic power, the whole will be 
shown to consist of an immense collection of exceedingly minute 
stars. 

These great telescopes, by the superior light they afford, 
display their peculiar powers in this department of astronomy, 
and those astronomers who, for the first time, have gazed at these 
sidereal pictures as seen in the " Leviathan" of Lord Rosse, have 
expressed, in glowing terms, their mingled delight and astonish- 
ment. The perfect forms, and strange but symmetrical config- 
urations, exhibited by these instruments, of nebulae that were 
before seen of irregular or fantastic shapes, afford grounds for 
believing that such irregularities are often if not always owing to 
the objects being but partly developed. Thus the Crab Nebula 
of Lord Rosse (Plate III. Fig. 4) had been long known as a faint, 
ill-defined nebula of an elliptical shape ; but the higher powers of 
that instrument exhibit the before concealed appendages which 
are essential to the completeness of the figure. The Whirlpool 
Nebula of Rosse, (Plate III. Fig. 5,) when seen in separate parts, 
exhibited no signs of order or symmetry ; but when viewed with 
the great Reflector, it develops the wonderful structure of a per- 
fect spiral. 

426. NebulaB were formerly divided into two classes, resolvable 
and irresolvable, the former term implying that the body was 
shown by the telescope to consist of stars, and the latter implying 
that the body is not composed of stars, but of a shining cloudy 
kind of matter diffused throughout the mass. Astronomers, at 
present, include all resolvable nebulae under the head of dusters, 
appropriating the term nebulae exclusively to such of these bodies 
as have never been resolved. The question whether this distinc- 
tion is not merely relative to the powers of the telescope, and 



CLUSTERS AND NEBULAE. 

whether, on the increase of these powers, this class of bodies 
would not all be resolved into stars, is not easily determined, since 
the same increase of telescopic power which converts existing 
nebulae into clusters, brings to light a greater number of those 
which are 'irresolvable. 

These remote objects of the universe occasionally exhibit traces 
of that regard to beauty which everywhere, in these nether 
worlds, characterizes the works of the Creator. In the Cross, a 
brilliant constellation of the southern hemisphere, for example, is a 
cluster surrounding the star Kappa Crucis, which consists of about 
110 stars from the seventh magnitude downwards, eight of the 
more conspicuous of which are colored with various shades of 
red, green, and blue, so as to give to the whole the appearance of 
a rich piece of jewelry. 

427. Nebulous stars are such as exhibit a sharp and brilliant 
star, surrounded by a disk or atmosphere of nebulous matter. 
These atmospheres, in some cases, present a circular, in others an 
oval figure ; and in certain instances, the nebula consists of a 
long, narrow, spindle-shaped ray, tapering away at both ends to 
points. Annular Nebula (Ring-shaped) -are among the rarest 
objects in the heavens. The most conspicuous of this class is in 
the Constellation Lyra, between the stars Beta and Gamma, about 
6 S. E. of Alpha Lyrae. This remarkable object is believed to 
be in fact a resolvable nebula or cluster, and yet the greatest 
powers of the telescope hitherto applied have only effected such 
changes as are regarded as giving signs of resolvability, but its 
perfect resolution has not been attained. Should it be achieved 
by an increased power of the instrument, astronomers look for a 
splendid coronet of stars, more glorious, perhaps, than any thing 
hitherto discovered in the starry heavens. 

Planetary Nebulce constitute another variety, and are very re- 
markable objects. They have, as their name imports, exactly the 
appearance of planets. Whatever may be their nature, they 
must be of enormous magnitude. One of them is to be found in 
the parallel of v Aquarii, and about 5m. preceding that star. Its 
apparent diameter is about 20". Another in the Constellation 
Andromeda, presents a visible disk of 12", perfectly defined and 



300 FIXED STARS. 

round. Granting these objects to be equally distant from us with 
the stars, their real dimensions must be such as, on the lowest 
computation, would fill the orbit of Uranus. It is no less evident 
that, if they be solid bodies, of a solar nature, the intrinsic splen- 
dor of their surfaces must be almost infinitely inferior to that of 
the sun. A circular portion of the sun's disk, subtending an 
angle of 20", would give a light equal to 100 full moons ; while 
the objects in question are hardly, if at all, discernible with the 
naked eye.* 

428. The Milky Way, or Galaxy, is a well-known luminous 
zone, encircling the sphere nearly in the direction of a great cir- 
cle. Near the Swan, in the northern sky, it is seen to be divided 
into two bands, which remain asunder for 150, and then reunite. 
The Galaxy owes its peculiar appearance to the blended light of 
myriads of small stars too minute to be individually recognized by 
the naked eye, but which are seen in their true character by a 
telescope of only moderate powers. Sir William Herschel esti- 
mated that, on one occasion, in forty-one minutes, no less than 
258,000 stars passed through the' small field of his telescope. f In 
approaching the border of the Milky Way, there is found a regu- 
lar but rapid increase in the number of stars, even before entering 
the limits of the luminous zone itself. Sir J. Herschel computes 
the whole number of stars in the Milky Way at five and a half 
millions, including such only as are visible in his twenty feet 
reflector. The Galaxy is itself supposed to be a nebula, of which 
our sun with its planets forms a constituent part ; and that it ap- 
pears so much greater than other nebulae only in consequence 
of our situation with respect to it, and its greater proximity to 
our system.J 



* Herschel. 

f Plate II. Fig. 1, exhibits a telescopic view of a part of the southern portion of 
the Milky Way. 

\ In the course of instruction given to the students of Yale College, topics of this 
kind are more fully discussed in lectures on astronomy. 



CHAPTER III. 

MOTIONS OP THE FIXED STARS DISTANCES NATURE. 

429. IN 1803, Sir William Herschel first determined and an- 
nounced to the world, that there exist among the stars separate 
systems, composed of two stars, revolving about each other in 
regular orbits. These he denominated Binary Stars, to distinguish 
them from other double stars where no such motion is detected, 
and whose proximity to each other may possibly arise from cas- 
ual juxtaposition, or from one being in the range of the other. 
At present, more than a hundred of the binary stars are known, 
and as the number of such revolutions known among the double 
stars is constantly increasing as the times of comparison increase, 
it may be anticipated that, in after ages, so large a proportion of 
all the double stars will be found to possess this character, as to 
authorize the belief that they universally consist of subordinate 
systems, of which the members have a revolution around a common 
center of gravity. The periodic times of the binary stars are 
very various. While some (as Hercules, and TJ Cor once) .complete 
their revolutions in 30 or 40 years, others (as 7 Virginis) re- 
quire more than 170, and others still (as 65 Piscium) take up the 
long period of 3000 years.* Their orbits are in general more 
eccentric than those of the planets. That of Gamma Virginis, 
including the relative positions of the two components from 1837 
to 1860, is figured on Plate II. as drawn by Mr. E. P. Mason, 
in 1840-f 



* Smyth's Cycle, I. 300. 

f Sir John Herschel had computed the orbit of y Virginis, and had given it at 
625 years. Mason, from a discussion of all the observations, published to the date 
of 1838, combined with his own of 1840, found that this period was too great, and 
assigned as the true period 111 years, which is now acknowledged by the highest 
authorities, and even by Herschel himself, to be nearly its real tune of revolution. 



302 FIXED STARS. 

430. The revolutions of the binary stars have assured us of 
this most interesting fact, that the law of gravitation extends to 
the fixed stars. Before these discoveries, we could not decide, 
except by a feeble analogy, that this law transcended the bounds 
of the solar system. Indeed, our belief of the fact rested more 
upon our idea of unity of design in all the works of the Creator, 
than upon any certain proof; but the revolution of one star 
around another in obedience to forces which must be similar to 
those that govern the solar system, establishes the grand conclu- 
sion, that the law of gravitation is truly the law of the material 
universe. 

We have the same evidence (says Sir John Herschel) of the 
revolutions of the binary stars about each other, that we have of 
those of Saturn and Uranus about the sun ; and the correspond- 
ence between their calculated and observed places in such elon- 
gated ellipses, must be admitted to carry with it a proof of the 
prevalence of the Newtonian law of gravity in their systems, of 
the very same nature and cogency as that of the calculated and 
observed places of comets round the center of our own system. 

But (he adds) it is not with the revolutions of bodies of a plan- 
etary or cometary nature round a solar center that we are now 
concerned; it is with that of sun around sun, each, perhaps, ac 
companied with its train of planets and their satellites, closely 
shrouded from our view by the splendor of their respective suns, 
and crowded into a space, bearing hardly a greater proportion to 
the enormous interval which separates them, than the distances 
of the satellites of our planets from their primaries, bear to their 
distances from the sun itself. 

431. Some of the fixed stars appear to have a PROPER MOTION, 
or a real motion in space. 

The apparent change of place in the stars arising from the pre- 
cession of the equinoxes, the nutation of the earth's axis, the 
diminution of the obliquity of the ecliptic, and the aberration of 
light, have been already mentioned ; but after all these corrections 
are made, changes of place still occur, which cannot result from 
any changes in the earth, but must arise from changes in the stars 
themselves. Such motions are called the proper motions of the 



MOTIONS OP THE FIXED STARS. 303 

stars. Nearly 2000 years ago, Hipparchus and Ptolemy made 
the most accurate determinations in their power of the relative 
situations of the stars, and their observations have been trans- 
mitted to us in Ptolemy's Almagest ; from which it appears that 
the stars retain at least very nearly the same places now as they 
did at that period. Still, the more accurate methods of modern 
astronomers, have brought to light minute changes in the places of 
certain stars which force upon us the conclusion, either that our 
solar system causes an apparent displacement of certain stars, by 
a motion of its own in space, or that they have themselves a proper 
motion. Possibly, indeed, both these causes may operate. 

. 

432. If the sun, and of course the earth which accompanies 
him, is actually in motion, the fact may become manifest from 
the apparent approach of the stars in the region which he is leav- 
ing, and the recession of those which lie in the part of the heav- 
ens towards which he is travelling. Were two groves of trees 
situated on a plain at some distance apart, and we should go 
from one to the other, the trees before us would gradually 
appear further and further asunder, while those we left behind 
would appear to approach each other. Some years since, Sir 
William Herschel supposed he had detected changes of this kind 
among two sets of stars in opposite points of the heavens, and an- 
nounced that the solar system was in motion towards a point in 
the Constellation Hercules.* As, for many years after this an- 
nouncement, other astronomers failed to find evidence of such a 
motion of the solar system, the doctrine was generally discredited, 
until, within a few years, new and very refined researches have 
been instituted by several of the most eminent astronomers, which 
have fully confirmed the observations of Herschel. The great 
Russian astronomer, Struve, by a comparison of the best observa- 
tions, finds the exact point towards which the solar system is mov- 
ing is in a line which joins the two stars * and f* Herculis,f a 
point which can be easily found on the celestial globe, and thence 
transferred tc the heavens. (Right ascension 259, declination 



* Phil. Trans. 1783, 1805, and 1806. 
f 6tudes d' Astronomic Stellaire, p. 108. 



304 FIXED STARS. 

34^.) The researches of the younger Struve have conducted 
him to the velocity with which the solar system is moving in 
space. For having found that the arc traversed by the sun in a 
year is 0".3392, if viewed at the mean distance of the stars of the 
first magnitude, and having previously ascertained that the mean 
parallax of the stars of this class amounts to 0".209, he infers that 
the space through which the sun moves annually is 154,000,000 
miles. Great as this space is, yet it may be remarked that it 
is only about one-fourth that traversed by the earth in its revo- 
lution around the sun. Within the comparatively short period 
during which these observations on the solar motion have been 
continued, the direction appears rectilinear ; but all analogy leads 
to the belief that it is in fact a motion of revolution, although on 
account of the immense size of the orbit, and, consequently, its 
small curvature, many years will be requisite in order to determine 
the deviation from the line of the tangent.* 

433. When we reflect on the immense distance of the stars, 
we may readily believe that they may be in fact in rapid motion, 
and yet appear quiescent ; as a distant ship, under full sail, ap- 
pears at rest, although actually moving at the rate of ten knots 
an hour. Thus we have seen above that a motion of the sun in 
space, as seen from the nearest fixed stars, would make it de- 
scribe an arc of only about one-third of a second annually, although 
traversing a space of 154 millions of miles. But a small change 
in the place of a star in a single year may, in a long series of 
years, accumulate to a very sensible amount. For example, the 
latitude's of the three bright stars, Sirius, Arcturus, and Aldebaran, 
were determined by Hipparchus 130 years before the Christian 
era, and their assigned places are transmitted to us in the Alma- 
gest of Ptolemy. About the year 1700, Dr. Halley found that 
these stars had, during the interval of nearly 2000 years, moved 
southerly through the spaces respectively of 37', 42', and 33'. 
The immense pains that have of late years been bestowed upon 
catalogues of the stars, and especially of particular portions of the 
heavens, with the view 'of furnishing, to after ages, the most ac- 

* Grant's Hist. Phys. Ast. 567. 



DISTANCES OF THE FIXED STARS. 305 

curate data for comparison, will enable future astronomers to 
study the proper motions of the stars with far greater advantages 
than the present generation enjoys. In most cases where a proper 
motion in certain stars has been suspected, its annual amount has 
been so small, that many years are required to assure us that the 
effect is not owing to some other than a real progressive motion 
in the stars themselves ; but in a few instances the fact is too 
obvious to admit of any doubt. A small star in the leg of the 
Great Bear has an annual motion away from the neighboring 
stars of 7", and the two stars 61 Cygni, which are nearly equal, 
have remained constantly at the same, or nearly at the same 
distance of 15" for at least fifty years past. Meanwhile they have 
shifted their local situation in the heavens, 4' 23", the annual 
proper motion of each star being 5."3, by which quantity this 
system is every year carried along in some unknown path, by a 
motion which for many centuries must be regarded as uniform 
and rectilinear. ^ A greater proportion of the double stars than of 
any other indicate proper motions, especially the binary stars, or 
those which have a revolution around each other. Among stars 
not double, and no way differing from the rest in any other ob- 
vious particular, ^ Cassiopeiae has a proper motion, amounting to 
nearly 4" annually ; and another obscure star has been recently 
found to have a motion of nearly 8".* 

434. DISTANCES OF THE FIXED STARS. 

It has long been considered one of the highest problems that can 
be proposed to the human mind, to measure the distance to any 
of the fixed stars. Nothing more, indeed, would be necessary 
han to determine its horizontal parallax ; but this is so exceed- 
uigly small, that, until recently, all efforts to measure it had proved 
unavailing. For all measurements relating to the distances of 
the sun and planets, the diameter of the earth furnishes the base 
line, (Art. 87.) The length of this line being known, and likewise 
the horizontal parallax of the body whose distance is sought, we 
readily obtain the distance by the solution of a right-angled tri- 
angle, (Art. 80, Fig. 6.) But any star viewed from the opposite 

* Herschel's Outlines, (Ed. 1851.) 
39 



306 FIXED STARS. 

sides of the earth, would appear from both stations to occupy pre- 
cisely the same situation in the celestial sphere, and of course it 
would exhibit no horizontal parallax. But astronomers have 
endeavored to find a parallax in some of the fixed stars, by taking 
the diameter of the earth's orbit as a base line. Yet even a 
change of position, amounting to 190 millions of miles, has, until 
within a few years, proved insufficient to alter the apparent 
place of a single fixed star, from which it was concluded that the 
fixed stars have not even any annual parallax ; or that the angle 
subtended by the semidiameter of the earth's orbit, at the nearest 
fixed star, is insensible, xhe errors to which instrumental meas- 
urements are subject, arising from defects of the instruments them- 
selves, from errors of refraction, of aberration, of precession, of 
nutation, and from imperfections of observation, are such, that the 
angular determinations of celestial arcs, it was supposed, could 
not be relied on to less than 1" ; and the change of place in any star 
that had been examined for parallax being less than one second 
when viewed at opposite extremities of the earth's orbit, the con- 
clusion was, that the parallax of the fixed stars, if any exist, is too 
minute ever to be measured by instruments. According to this, 
the diameter of the earth's orbit, when viewed from the nearest 
fixed star, would be insensible ; the spider-line of the telescope 
would more than cover it. Fig. 80. 

Taking, however, the annual parallax at 1", let ab (Fig. 
80) represent the radius of the earth's orbit, and c a fixed 
star, the angle at c being 1", and the angle at b a right 
angle ; then, 

Sin. 1" : Rad. : : 1 : 200,000, nearly. 

Hence the hypothenuse of a triangle whose vertical 
angle is 1", is about 200,000 times the base; conse- 
quently, the distance in question must exceed 95,000,000 x 
200,000=190,000,000X100,000, or one hundred thou- 
sand times one hundred and ninety millions of miles. Of 
a distance so vast we can form no adequate conceptions, 
and attempt to measure it only by the time that light 
(which moves more than 192,000 miles per second) would 
take to traverse it. Now, 

192,000 : Is..: 19,000,000,000,000 : 3.1 vears 



DISTANCES OF THE FIXED STARS. 307 

435. After many fruitless and delusory efforts to measure the 
immense interval that separates us from the fixed stars, the great 
Prussian astronomer, Bessel, in the year 1838, determined this 
interesting and important element, by observations on a double 
star in the Swan, (61 Cygni.) This star was selected for the 
following reasons : first, it was known to have a great proper 
motion, (Art. 4i3,) indicating a comparatively great prox- 
imity to our system ; secondly, situated as it is among the 
circumpolar stars, observations could be made upon it nearly 
every night in the year ; and, thirdly, the great number of small 
stars in the immediate neighborhood, furnished the opportunity 
of selecting favorable stationary points from which (inasmuch as 
these more remote objects might be considered as entirely devoid 
of parallax) any changes of place in the nearer, in consequence 
of an annual parallax, might be readily estimated. By observa- 
tions of the last degree of refinement, conducted for a period of 
several years, a parallax was decisively indicated, amounting to 
about one-third of a second ; or, more exactly, to 0."3483, imply- 
ing a distance of 592,200 times the mean distance of the earth 
from the sun, or a space which it would take light, moving at the 
rate of twelve millions of miles per minute, nine and a quartet 
years to traverse. To form some familiar notions of this distance, 
let us suppose a railway-car to travel night and day, at the rate 
of twenty miles an hour : we should find it would take it about 547 
years to reach the sun ; but to reach 61 Cygni would require 
324,000,000 of years. 

The observations of Bessel enabled him to estimate also the 
period of revolution of the two stars composing the binary sys- 
tem of 61 Cygni, and the dimensions of the orbit, and he found 
the periodic time about 540 years, and the length of the orbit 
about two and a half times that of Uranus. Knowing also the 
distance of this star, we can now determine from its proper mo- 
tion (five seconds a year) the velocity of its motion : this is found 
to be about forty-four miles per second more than double that 
of the earth in its orbit amounting to about one thousand mil- 
lions of miles per annum. 

On account of the smallness of the supposed parallax thus 
found, it would not be unreasonable still to entertain a lingering 



308 FIXED STARS. 

suspicion, that it is nothing more than the unavoidable imperfec 
tion of instrumental measurements, as proved to be the case in 
previous attempts to find the same element ; but the most satis- 
factory evidence which the world can have that such is not the 
fact in the present instance, but that the parallax is truly found, 
is that the most celebrated astronomers of the age, after rigorous 
scrutiny, have acknowledged the reality and soundness of the de- 
termination. Our confidence that the parallax of 61 Cygni was 
truly determined by Bessel, is strengthened by the fact that a sep- 
arate determination recently made by Peters at the Pulkova Ob- 
servatory, gives almost precisely the same result, that of Bessel 
being 0. // 348, and that of Peters 0."349. In the case of several stars 
still more distant, the parallax has been found, with more or less 
probability, but with sufficient to command the general confidence 
of astronomers. Thus, the parallax of Arcturus, Alpha Lyrse, 
and Polaris, were also found by Peters to be respectively 0/'127, 
0.' 123, O."067, that of the Pole-star being only one-fifth as great 
as that of 61 Cygni; and, consequently, if light would require 
9j years to come from that star, it would require more than 46 
years to come to us from the Pole-star. A star in the southern 
hemisphere, (a Centauri,) indicates a parallax of about 1", and 
hence appears at present the nearest of the fixed stars. 



436. NATURE OF THE STARS. 

The stars are bodies' greater than our earth. If this were 
not the case they could not be visible at such an immense dis- 
tance. Dr. Wollaston, a distinguished English philosopher, 
attempted to estimate the magnitudes of certain of the fixed 
stars from the light which they afford. By means of an accu- 
rate photometer (an instrument for measuring the relative inten- 
sities of light) he compared the light of Sirius with that of the 
sun. He next inquired how far the sun must be removed from 
us in order to appear no brighter than Sirius. He found the dis- 
tance to be 141,400 times its present distance. But Sirius is more 
than 200,000 times as far off as the sun, (Art. 434.) Hence he 
inferred that, upon the lowest computation, Sirius must actually 
give out twice as much light as the sun ; or that, in point of 



NATURE OF THE STARS. 309 

splendor, Sirius must be at least equal to two suns. Indeed, he 
has rendered it probable that the light of Sirius is equal to four- 
teen suns. 

437. The fixed stars are suns. We have already seen that 
they are large bodies ; that they are immensely further off than 
the furthest planet ; that they shine by their own light, as is evi- 
dent by the nature of the light as tested by polarization : in short, 
that their appearance is, in all respects, the same as the sun would 
exhibit if removed to the region of the stars. Hence we infer 
that they are bodies of the same kind with the sun. 

438. We are justified therefore by a sound analogy, in con- 
cluding that the stars were made for the same end as the sun, 
namely, as the centers of attraction to other planetary worlds, to 
which they severally dispense light and heat. Although the starry 
heavens present, in a clear night, a spectacle of ineffable gran- 
deur and beauty, yet it must be admitted that the chief purpose 
of the stars could not have been to adorn the night, since by far 
the greatest part of them are wholly invisible to the naked eye ; 
nor as landmarks to the navigator, for only a very small propor- 
tion of them are adapted for this purpose ; nor, finally, to influence 
the earth by their attractions, since their distance renders such 
an effect entirely insensible. If they are suns, and if they exert 
no important agencies upon our world, but are bodies evidently 
adapted to the same purpose as our sun, then it is as rational to 
suppose that they were made to give light and heat, as that the 
eye was made for seeing and the ear for hearing. It is obvious 
to inquire next, to what they dispense these gifts if not to plan- 
etary worlds ; and why to planetary worlds, if not for the use of 
percipient beings ? We are thus led, almost inevitably, to the 
idea of a Plurality of Worlds ; and the conclusion is forced upon 
us, that the spot which the Creator has assigned to us is but a 
humble province of his boundless empire.* 

* See this argument, in its full extent, in Dick's Celestial Scenery. 



CHAPTER IV. 

OP THE SYSTEM OF THE WORLD. 

439. The arrangement of all the bodies that compose the ma- 
terial universe, and their relations to each other, constitutes the 
System of the World. 

It is otherwise called the Mechanism of the Heavens ; and in- 
deed in the System of the World, we figure to ourselves a machine, 
all the parts of which have a mutual dependence, and conspire to 
one great end. " The machines that are first invented (says 
Adam Smith) to perform any particular movement, are always 
the most complex ; and succeeding artists generally discover that 
with fewer wheels and with fewer principles of motion than had 
originally been employed, the same effects may be more easily 
produced. The first systems, in the same manner, are always the 
most complex ; and a particular connecting chain or principle is 
generally thought necessary to unite every two seemingly dis- 
jointed appearances ; but it often happens, that one great connect- 
ing principle is afterwards found to be sufficient to bind together 
all the discordant phenomena that occur in a whole species of 
things." This remark is strikingly applicable to the origin and 
progress of systems of astronomy. 

440. From the visionary notions which are generally under- 
stood to have been entertained on this subject by the ancients, 
we are apt to imagine that they knew less than they actually did 
of the truths of astronomy. But Pythagoras, who lived 500 years 
before the Christian era, was acquainted with many important 
facts in our science, and entertained many opinions respecting 
the System of the World which are now held to be true. Among 
other things well known to Pythagoras were the following : 

1. The principal Constellations. These had begun to be formed 
in the earliest ages of the world. Several of them bearing the 
same names as at present are mentioned in the writings of Hesiod 



SYSTEM OF THE WORLD. 311 

and Homer ; and the " sweet influences of the Pleiades" and the 
"bands of Orion," are beautifully alluded to in the Book of Job. 

2. Eclipses. Pythagoras knew both the causes of eclipses and 
how to predict them ;* not indeed in the accurate manner now 
employed, but by means of the Saros, (Art. 233.) 

3. Pythagoras had divined the true system of the world, hold- 
ing that the sun, and not the earth, (as was generally held by the 
ancients, even for many years after Pythagoras,) is the center 
around which all the planets revolve, and that the stars are so 
many suns, each the center of a system like our own.f Among 
lesser things, he knew that the earth is round ; that its surface 
is naturally divided into five zones ; and that the ecliptic is in- 
clined to the equator. He also held that the earth revolves daily 
on its axis, and yearly around the sun ; that the galaxy is an as- 
semblage of small stars ; and that it is the same luminary, namely, 
Venus, that constitutes both the morning and the evening star, 
whereas all the ancients before him had supposed that each was 
a separate planet, and accordingly the morning star was called 
Lucifer, and the evening star Hesperus. J He held also that the 
planets were inhabited, and even went so far as to calculate the 
size of some of the animals in the moon. Pythagoras was so great 
an enthusiast in music, that he not only assigned to it a conspicuous 
place in his system of education, but even supposed the heavenly 
bodies themselves to be arranged at distances corresponding to 
the diatonic scale, and imagined them to pursue their sublime 
march to notes created by their own harmonious movements, 
called the " music of the spheres ;" but he maintained that this 
celestial concert, though loud and grand, is not audible to the 
feeble organs of man, but only to the gods. 

441. With few exceptions, however, the opinions of Pythago- 
ras on the System of the World, were founded in truth. Yet they 
were rejected by Aristotle and by most succeeding astronomers 
down to the time of Copernicus, and in their place was substituted 

* Long's Astronomy, ii. 671. 

f Library of Useful Knowledge, History of Astronomy. 

\ Long's Ast. ii. 678. Ed. Encyclopaedia. 



312 SYSTEM OF THE WORLD. 

the doctrine of Crystalline Spheres, first taught by Eudoxus. A c- 
cording to this system, the heavenly bodies are set like gems in 
hollow solid orbs, composed of crystal so pellucid that no anterior 
orb obstructs in the least the view of any of the orbs that lie behind 
it. The sun and the planets have each its separate orb ; but the 
fixed stars are all set in the same grand orb ; and beyond this is 
another still, the Primum Mobile, which revolves daily from east 
to west, and carries along with it all the other orbs. Above the 
whole, spreads the Grand Empyrean, or third heavens, the abode 
of perpetual serenity.* 

To account for the planetary motions, it was supposed that each 
of the planetary orbs, as well as that of the sun, has a motion of its 
own eastward, while it partakes of the common diurnal motion of 
the starry sphere. Aristotle taught that these motions are effected 
by a tutelary genius of each planet, residing in it, and directing 
its motions, as the mind of man directs his motions. 

442. On coming down to the time of Hipparchus, who flourished 
about 150 years before the Christian era, we meet with astrono- 
mers who acquired far more accurate knowledge of the celestial 
motions. Hipparchus was in possession of instruments for meas- 
uring angles, and knew, how to resolve spherical triangles. He 
ascertained the length of the year within 6m. of the truth. He 
discovered the eccentricity of the solar orbit, (although he sup- 
posed the sun actually to move uniformly in a circle, but the earth 
to be placed out of the center,) and the positions of the sun's 
apogee and perigee. He formed very accurate estimates of the 
obliquity of the ecliptic and of the precession of the equinoxes. 
He computed the exact period of the synodic revolution of the 
moon, and the inclination of the lunar orbit ; discovered the mo- 
tion of her node and of her line of apsides ; and made the first 
attempts to ascertain the horizontal parallaxes of the sun and moon. 

Such was the state of astronomical knowledge when Ptolemy 
wrote the Almagest, in which he has transmitted to us an en- 
cyclopaedia of the astronomy of the ancients. 



* Long's Ast. ii. 640 Robinson's Mech. Phil. ii. 83- Gregory's Ast. 132 Play- 
fair's Dissertations, 118. 



THE PTOLEMAIC SYSTEM. 313 

443. The systems of the world which have been most celebrated 
are three the Ptolemaic, the Tychonic, and the Copernican. 
We shall conclude this part of our work with a concise statement 
and discussion of each of these systems of the Mechanism of 
the Heavens. 

THE PTOLEMAIC SYSTEM. 

444. The doctrines of the Ptolemaic System were not originated 
by Ptolemy, but being digested by him out of materials furnished 
by various hands, it has come down to us under the sanction of 
his name. 

According to this system, the earth is the center of the uni- 
verse, and all the heavenly bodies daily revolve around it from 
east to west. In order to explain the planetary motions, Ptolemy 
had recourse to deferents and epicycles, an explanation devised 
by Apollonius, one of the greatest geometers of antiquity.* He 
conceived that, in the circumference of a circle, having the earth 
for its center, there moves the center of another circle, in the 
circumference of which the planet actually revolves. The circle 
surrounding the earth was called the deferent, while the smaller 
circle, whose center was always in the periphery of the deferent, 
was called the epicycle. The motion in each was supposed to be 
uniform. Lastly, it was conceived that the motion of the center 
of the epicycle in the circumference of the deferent, and of the 
deferent itself, are in opposite directions, the first being towards 

the east, and the second towards the west. 
, \ 

445. But these views will be better understood from a diagram. 
Therefore, let ABC (Fig. 81) represent the deferent, E being the 
earth a little out of the center. Let abc represent the epicycle y 
having its center at v, on the periphery of the deferent. Con- 
ceive the circumference of the deferent to be carried about the 
earth every twenty- four hours in the order of the letters ; and at 
the same time, let the center v of the epicycle abed, have a slow 
motion in the opposite direction, and let a body revolve in this 



* Playfair, Dissertation Second, 119. 
40 



314 



SYSTEM OF THE WORLD. 



circle in the direction abed. Then it will be seen that the body 
would actually describe the looped curves klmnop ; that it would 
appear stationary at / and m, and at n and o ; that its motion 
would be direct from k to /, and then retrograde from I to m ; 
direct again from m to n, and retrograde from n to o. Thus, 
suppose Mercury to be situated at b in its epicycle. By the rev- 
olution of the deferent, it would be carried along with the other 




heavenly bodies around the earth from left to right, every twenty- 
four hours ; but, meanwhile, the center of the epicycle shifting 
its place slowly from right to left, while Mercury was moving 
from b to c, c itself would change its place to r, and therefore the 
path of the planet would be in the cycloidal arc br. Again, while 
Mercury was passing through cda, the point c would be still mov- 
ing eastward, which would have the effect apparently to compress 
the lower half of the epicycle into the looped curve nor ; and as 
on this side the motion in the epicycle is in the same direction 
with that of the deferent, but at a slower rate, the apparent path 
is much shorter than where, as on the other side, the two motions 
conspire. 



THE TYCHONIC SYSTEM. 315 

446. Such a deferent and epicycle may be devised for each 
planet as will fully explain all its ordinary motions ; but it is in- 
consistent with the phases of Mercury and Venus, which being 
between us and the sun on both sides of the epicycle, would pre- 
sent their dark sides towards us in both these positions, whereas 
at one of the conjunctions they are seen to shine with full face.* 
It is moreover absurd to speak of a geometrical center, which has 
no bodily existence, moving around the earth on the circumference 
of another circle ; and hence some suppose that the ancients 
merely assumed this hypothesis as affording a convenient geo- 
metrical representation of the phenomena, a diagram simply, 
without conceiving the system to have any real existence in 
nature. 

447. The objections to the Ptolemaic system, in general, are the 
following : First, it is a mere hypothesis, having no evidence in 
its favor, except that it explains the phenomena. This evidence 
is insufficient of itself, since it frequently happens that each of 
two hypotheses, directly opposite to each other, will explain all 
the known phenomena. But the Ptolemaic system does not even 
do this, as it is "inconsistent with the phases of Mercury and 
Venus, as already observed. Secondly, now that we are ac- 
quainted with the distances of the remoter planets, and especially 
of the fixed stars, the swiftness of motion implied in a daily rev- 
olution of the starry firmament around the earth, renders such a 
motion wholly incredible. Thirdly, the centrifugal force that 
would be generated in these bodies, especially in the sun, renders 
it impossible that they can continue to revolve around the earth 
as a center. 

These reasons are sufficient to show the absurdities of the 
Ptolemaic System of the World. 

THE TYCHONIC SYSTEM. 

448. Tycho Brahe, like Ptolemy, placed the earth in the center 
of the universe, and accounted for the diurnal motions in the same 

* Vince's Complete System, i 90. 



316 SYSTEM OF THE WORLD. 

manner as Ptolemy had done, namely, by an actual revolution of 
the whole host of heaven around the earth every twenty-four 
hours. But he rejected the scheme of deferents and epicycles, 
and held that the moon revolves about the earth as the center of 
her motions ; that the sun, and not the earth, is the center of the 
planetary motions ; and that the sun, accompanied by the planets, 
moves around the earth once a year, somewhat in the manner 
that we now conceive of Jupiter and his satellites as revolving 
around the sun. The system of Tycho serves to explain all the 
common phenomena of the planetary motions, but it is encum- 
bered with the same objections as those that have been men- 
tioned as resting against the Ptolemaic system, namely, that it is 
a mere hypothesis ; that it implies an incredible swiftness in the 
diurnal motions ; and that it is inconsistent with the known laws 
of universal gravitation. But if the heavens do not revolve, the 
earth must, and this brings us to the system of Copernicus. 



THE COPERNICAN SYSTEM. 

449. Copernicus was born at Thorn, in Prussia, in 1473. The 
system that bears his name was the fruit of forty years of intense 
study and meditation upon the celestial motions. As already 
mentioned, (Art. 6,) it maintains (1) That the apparent diurnal 
motions of the heavenly bodies, from east to west, is owing to the 
real revolution of the earth on its own axis from west to east ; 
and (2) That the sun is the center around which the earth and 
planets all revolve from west to east. It rests on the following 
arguments : 

First, the earth revolves on its own axis. 

1. Because this supposition is vastly more simple. 

2. It is agreeable to analogy, since all the other planets that 
afford any means of determining the question, are seen to revolve 
on their axes. 

3. The spheroidal figure of the earth is the figure of equilib- 
rium, that results from a revolution on its axis. 

4.. The diminished weight of bodies at the equator, indicates a 
centrifugal force arising from such a revolution. 

5. Bodies let fall from a high eminence, fall eastward of their 



THE COPERNICAN SYSTEM. 317 

base, indicating that when further from the center of the earth 
they were subject to a greater velocity, which, in consequence of 
their inertia, they do not entirely lose in descending to the lower 
level.* 

Secondly, the planets, including the earth, revolve about the sun. 

1. The phases of Mercury and Venus are precisely such as 
would .result from their circulating around the sun in orbits within 
that of k the earth; but they are never seen in opposition, as they 
would be if they circulated around the earth. 

2. The superior planets do indeed revolve around the earth ; 
but they also revolve around the sun, as is evident from their 
phases and from the known dimensions of their orbits ; and that 
the sun, and not the earth, is the center of their motions, is in- 
ferred from the greater symmetry of their motions as referred to 
the sun than as referred to the earth, and especially from the laws 
of gravitation, which forbid our supposing that bodies so mucl 
larger than the earth, as some of these bodies are, can circulate 
permanently around the earth, the latter remaining all the while 
at rest. 

3. The annual motion of the earth itself is indicated also by the 
most conclusive arguments. For, first, since all the planets with 
their satellites, and the comets, revolve about the sun, analogy 
leads us to infer the same respecting the earth and its satellite. 
Secondly, the motions of the satellites, as those of Jupiter and 
Saturn, indicate that it is a law of the solar system that the 
smaller bodies revolve about the larger. Thirdly, the direction 
of the periodical meteors of November, which, in a majority of 
cases, is from east to west, indicates the motion of the earth from 
west to east. Lastly, the aberration of light affords a sensible 
proof of the motion of the earth, since that phenomenon indicates 
both a progressive motion of light, and a motion of the earth from 
west to east. (Art. 195.) 

450. It only remains to inquire whether there subsist higher 
orders of relations between the stars themselves. The assem- 
blage of bodies in clusters, as in the Pleiades, and still more, as in 

* Biot. 









-" *'* T ;-<*fc<5^ 



j*i ? 
^ - 



' . 



t>LATE II. 

NEBULA AND DOUBLE STARS. 




1. Castor. 2. y Leonis. 3. 39 Drac. 4. A Oph. 5. 11 Monoc. 6. cCancri 




Revolutions of y Virginis. 



n 

1837. 1838. 1839, 1840. 1845. 1850. 1860. Orbit. 



VLATE ILL 

CLUSTERS AND NEBULA, 





UNIVERSITY OF CALIFORNIA LIBRARY 



